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instiki_ncatlab.org.atom.xml (502820B)
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1 <feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en">
2 <title>nLab</title>
3 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/HomePage"/>
4 <link rel="self" href="https://ncatlab.org/nlab/atom_with_content"/>
5 <updated>2021-07-02T09:22:44Z</updated>
6 <id>tag:ncatlab.org,2008-11-28:nLab</id>
7 <subtitle>An Instiki Wiki</subtitle>
8 <generator uri="http://golem.ph.utexas.edu/instiki/show/HomePage" version="0.19.7(MML+)">Instiki</generator>
9 <entry>
10 <title type="html">Urs Frauenfelder</title>
11 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Urs+Frauenfelder"/>
12 <updated>2021-07-02T09:22:44Z</updated>
13 <published>2021-07-02T09:22:46Z</published>
14 <id>tag:ncatlab.org,2021-07-02:nLab,Urs+Frauenfelder</id>
15 <author>
16 <name>Urs Schreiber</name>
17 </author>
18 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Urs+Frauenfelder">
19 <div xmlns="http://www.w3.org/1999/xhtml">
20 <ul>
21 <li>
22 <p><a href='https://www.uni-augsburg.de/en/fakultaet/mntf/math/prof/geom/frauenfelder/'>Institute page</a></p>
23 </li>
24
25 <li>
26 <p><a href='https://www.genealogy.math.ndsu.nodak.edu/id.php?id=120240'>MathematicsGenealogy page</a></p>
27 </li>
28 </ul>
29
30 <h2 id='selected_writings'>Selected writings</h2>
31
32 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cyclic loop spaces</a>:</p>
33
34 <ul>
35 <li><a class='existingWikiWord' href='/nlab/show/Urs+Frauenfelder'>Urs Frauenfelder</a>, <em>Dihedral homology and the moon</em>, J. Fixed Point Theory Appl. <strong>14</strong> (2013) 55–69 (<a href='https://arxiv.org/abs/1204.4549'>arXiv:1204.4549</a>, <a href='https://doi.org/10.1007/s11784-013-0146-z'>doi:10.1007/s11784-013-0146-z</a>)</li>
36 </ul>
37
38 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p> </div>
39 </content>
40 </entry>
41 <entry>
42 <title type="html">cyclic loop space</title>
43 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/cyclic+loop+space"/>
44 <updated>2021-07-02T09:20:18Z</updated>
45 <published>2017-02-14T09:56:17Z</published>
46 <id>tag:ncatlab.org,2017-02-14:nLab,cyclic+loop+space</id>
47 <author>
48 <name>Urs Schreiber</name>
49 </author>
50 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/cyclic+loop+space">
51 <div xmlns="http://www.w3.org/1999/xhtml">
52 <div class='rightHandSide'>
53 <div class='toc clickDown' tabindex='0'>
54 <h3 id='context'>Context</h3>
55
56 <h4 id='topology'>Topology</h4>
57
58 <div class='hide'>
59 <p><strong><a class='existingWikiWord' href='/nlab/show/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/point-free+topology'>point-free topology</a>)</p>
60
61 <p>see also <em><a class='existingWikiWord' href='/nlab/show/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></em></p>
62
63 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology'>Introduction</a></p>
64
65 <p><strong>Basic concepts</strong></p>
66
67 <ul>
68 <li>
69 <p><a class='existingWikiWord' href='/nlab/show/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/neighborhood'>neighbourhood</a></p>
70 </li>
71
72 <li>
73 <p><a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/locale'>locale</a></p>
74 </li>
75
76 <li>
77 <p><a class='existingWikiWord' href='/nlab/show/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/neighborhood+base'>neighbourhood base</a></p>
78 </li>
79
80 <li>
81 <p><a class='existingWikiWord' href='/nlab/show/finer+topology'>finer/coarser topology</a></p>
82 </li>
83
84 <li>
85 <p><a class='existingWikiWord' href='/nlab/show/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a></p>
86 </li>
87
88 <li>
89 <p><a class='existingWikiWord' href='/nlab/show/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/sober+topological+space'>sobriety</a></p>
90 </li>
91
92 <li>
93 <p><a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/homeomorphism'>homeomorphism</a></p>
94 </li>
95
96 <li>
97 <p><a class='existingWikiWord' href='/nlab/show/uniformly+continuous+map'>uniformly continuous function</a></p>
98 </li>
99
100 <li>
101 <p><a class='existingWikiWord' href='/nlab/show/embedding+of+topological+spaces'>embedding</a></p>
102 </li>
103
104 <li>
105 <p><a class='existingWikiWord' href='/nlab/show/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/closed+map'>closed map</a></p>
106 </li>
107
108 <li>
109 <p><a class='existingWikiWord' href='/nlab/show/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/filter'>filter</a></p>
110 </li>
111
112 <li>
113 <p><a class='existingWikiWord' href='/nlab/show/convergence'>convergence</a></p>
114 </li>
115
116 <li>
117 <p><a class='existingWikiWord' href='/nlab/show/category'>category</a> <a class='existingWikiWord' href='/nlab/show/Top'>Top</a></p>
118
119 <ul>
120 <li><a class='existingWikiWord' href='/nlab/show/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li>
121 </ul>
122 </li>
123 </ul>
124
125 <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p>
126
127 <ul>
128 <li>
129 <p><a class='existingWikiWord' href='/nlab/show/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/weak+topology'>final topology</a></p>
130 </li>
131
132 <li>
133 <p><a class='existingWikiWord' href='/nlab/show/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient space</a>,</p>
134 </li>
135
136 <li>
137 <p>fiber space, <a class='existingWikiWord' href='/nlab/show/space+attachment'>space attachment</a></p>
138 </li>
139
140 <li>
141 <p><a class='existingWikiWord' href='/nlab/show/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/disjoint+union+topological+space'>disjoint union space</a></p>
142 </li>
143
144 <li>
145 <p><a class='existingWikiWord' href='/nlab/show/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/cocylinder'>mapping cocylinder</a></p>
146 </li>
147
148 <li>
149 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
150 </li>
151
152 <li>
153 <p><a class='existingWikiWord' href='/nlab/show/mapping+telescope'>mapping telescope</a></p>
154 </li>
155
156 <li>
157 <p><a class='existingWikiWord' href='/nlab/show/colimits+of+normal+spaces'>colimits of normal spaces</a></p>
158 </li>
159 </ul>
160
161 <p><strong><a class='existingWikiWord' href='/nlab/show/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p>
162
163 <ul>
164 <li>
165 <p><a class='existingWikiWord' href='/nlab/show/nice+topological+space'>nice topological space</a></p>
166 </li>
167
168 <li>
169 <p><a class='existingWikiWord' href='/nlab/show/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/metrisable+topological+space'>metrisable space</a></p>
170 </li>
171
172 <li>
173 <p><a class='existingWikiWord' href='/nlab/show/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/normal+space'>normal space</a></p>
174 </li>
175
176 <li>
177 <p><a class='existingWikiWord' href='/nlab/show/sober+topological+space'>sober space</a></p>
178 </li>
179
180 <li>
181 <p><a class='existingWikiWord' href='/nlab/show/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/proper+map'>proper map</a></p>
182
183 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/strongly+compact+topological+space'>strongly compact</a></p>
184 </li>
185
186 <li>
187 <p><a class='existingWikiWord' href='/nlab/show/compactly+generated+topological+space'>compactly generated space</a></p>
188 </li>
189
190 <li>
191 <p><a class='existingWikiWord' href='/nlab/show/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/first-countable+space'>first-countable space</a></p>
192 </li>
193
194 <li>
195 <p><a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/locally+contractible+space'>locally contractible space</a></p>
196 </li>
197
198 <li>
199 <p><a class='existingWikiWord' href='/nlab/show/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/locally+connected+topological+space'>locally connected space</a></p>
200 </li>
201
202 <li>
203 <p><a class='existingWikiWord' href='/nlab/show/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p>
204 </li>
205
206 <li>
207 <p><a class='existingWikiWord' href='/nlab/show/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/CW+complex'>CW-complex</a></p>
208 </li>
209
210 <li>
211 <p><a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed space</a></p>
212 </li>
213
214 <li>
215 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/Hilbert+space'>Hilbert space</a></p>
216 </li>
217
218 <li>
219 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topological group</a></p>
220 </li>
221
222 <li>
223 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/topological+K-theory'>topological K-theory</a></p>
224 </li>
225
226 <li>
227 <p><a class='existingWikiWord' href='/nlab/show/topological+manifold'>topological manifold</a></p>
228 </li>
229 </ul>
230
231 <p><strong>Examples</strong></p>
232
233 <ul>
234 <li>
235 <p><a class='existingWikiWord' href='/nlab/show/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/point+space'>point space</a></p>
236 </li>
237
238 <li>
239 <p><a class='existingWikiWord' href='/nlab/show/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/codiscrete+space'>codiscrete space</a></p>
240 </li>
241
242 <li>
243 <p><a class='existingWikiWord' href='/nlab/show/Sierpinski+space'>Sierpinski space</a></p>
244 </li>
245
246 <li>
247 <p><a class='existingWikiWord' href='/nlab/show/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/Scott+topology'>Scott topology</a></p>
248 </li>
249
250 <li>
251 <p><a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclidean space</a></p>
252
253 <ul>
254 <li><a class='existingWikiWord' href='/nlab/show/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/plane'>plane</a></li>
255 </ul>
256 </li>
257
258 <li>
259 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/cone'>cone</a></p>
260 </li>
261
262 <li>
263 <p><a class='existingWikiWord' href='/nlab/show/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/ball'>ball</a></p>
264 </li>
265
266 <li>
267 <p><a class='existingWikiWord' href='/nlab/show/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/M%C3%B6bius+strip'>Moebius strip</a></p>
268 </li>
269
270 <li>
271 <p><a class='existingWikiWord' href='/nlab/show/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/polyhedron'>polyhedron</a></p>
272 </li>
273
274 <li>
275 <p><a class='existingWikiWord' href='/nlab/show/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/complex+projective+space'>complex</a>)</p>
276 </li>
277
278 <li>
279 <p><a class='existingWikiWord' href='/nlab/show/classifying+space'>classifying space</a></p>
280 </li>
281
282 <li>
283 <p><a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration space</a></p>
284 </li>
285
286 <li>
287 <p><a class='existingWikiWord' href='/nlab/show/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/loop'>loop</a></p>
288 </li>
289
290 <li>
291 <p><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/topology+of+uniform+convergence'>topology of uniform convergence</a></p>
292
293 <ul>
294 <li><a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/path+space'>path space</a></li>
295 </ul>
296 </li>
297
298 <li>
299 <p><a class='existingWikiWord' href='/nlab/show/Zariski+topology'>Zariski topology</a></p>
300 </li>
301
302 <li>
303 <p><a class='existingWikiWord' href='/nlab/show/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/Mandelbrot+set'>Mandelbrot space</a></p>
304 </li>
305
306 <li>
307 <p><a class='existingWikiWord' href='/nlab/show/Peano+curve'>Peano curve</a></p>
308 </li>
309
310 <li>
311 <p><a class='existingWikiWord' href='/nlab/show/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/Sorgenfrey+line'>Sorgenfrey line</a></p>
312 </li>
313
314 <li>
315 <p><a class='existingWikiWord' href='/nlab/show/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/Dowker+space'>Dowker space</a></p>
316 </li>
317
318 <li>
319 <p><a class='existingWikiWord' href='/nlab/show/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/Hawaiian+earring+space'>Hawaiian earring space</a></p>
320 </li>
321 </ul>
322
323 <p><strong>Basic statements</strong></p>
324
325 <ul>
326 <li>
327 <p><a class='existingWikiWord' href='/nlab/show/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p>
328 </li>
329
330 <li>
331 <p><a class='existingWikiWord' href='/nlab/show/schemes+are+sober'>schemes are sober</a></p>
332 </li>
333
334 <li>
335 <p><a class='existingWikiWord' href='/nlab/show/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p>
336 </li>
337
338 <li>
339 <p><a class='existingWikiWord' href='/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p>
340 </li>
341
342 <li>
343 <p><a class='existingWikiWord' href='/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p>
344 </li>
345
346 <li>
347 <p><a class='existingWikiWord' href='/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p>
348 </li>
349
350 <li>
351 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net'>compact spaces equivalently have converging subnet of every net</a></p>
352
353 <ul>
354 <li>
355 <p><a class='existingWikiWord' href='/nlab/show/Lebesgue+number+lemma'>Lebesgue number lemma</a></p>
356 </li>
357
358 <li>
359 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p>
360 </li>
361
362 <li>
363 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net'>compact spaces equivalently have converging subnet of every net</a></p>
364 </li>
365
366 <li>
367 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p>
368 </li>
369 </ul>
370 </li>
371
372 <li>
373 <p><a class='existingWikiWord' href='/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p>
374 </li>
375
376 <li>
377 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p>
378 </li>
379
380 <li>
381 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p>
382 </li>
383
384 <li>
385 <p><a class='existingWikiWord' href='/nlab/show/closed+injections+are+embeddings'>closed injections are embeddings</a></p>
386 </li>
387
388 <li>
389 <p><a class='existingWikiWord' href='/nlab/show/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p>
390 </li>
391
392 <li>
393 <p><a class='existingWikiWord' href='/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p>
394 </li>
395
396 <li>
397 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p>
398 </li>
399
400 <li>
401 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p>
402 </li>
403
404 <li>
405 <p><a class='existingWikiWord' href='/nlab/show/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p>
406 </li>
407
408 <li>
409 <p><a class='existingWikiWord' href='/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p>
410 </li>
411 </ul>
412
413 <p><strong>Theorems</strong></p>
414
415 <ul>
416 <li>
417 <p><a class='existingWikiWord' href='/nlab/show/Urysohn%27s+lemma'>Urysohn&#39;s lemma</a></p>
418 </li>
419
420 <li>
421 <p><a class='existingWikiWord' href='/nlab/show/Tietze+extension+theorem'>Tietze extension theorem</a></p>
422 </li>
423
424 <li>
425 <p><a class='existingWikiWord' href='/nlab/show/Tychonoff+theorem'>Tychonoff theorem</a></p>
426 </li>
427
428 <li>
429 <p><a class='existingWikiWord' href='/nlab/show/tube+lemma'>tube lemma</a></p>
430 </li>
431
432 <li>
433 <p><a class='existingWikiWord' href='/nlab/show/Michael%27s+theorem'>Michael&#39;s theorem</a></p>
434 </li>
435
436 <li>
437 <p><a class='existingWikiWord' href='/nlab/show/Brouwer%27s+fixed+point+theorem'>Brouwer&#39;s fixed point theorem</a></p>
438 </li>
439
440 <li>
441 <p><a class='existingWikiWord' href='/nlab/show/topological+invariance+of+dimension'>topological invariance of dimension</a></p>
442 </li>
443
444 <li>
445 <p><a class='existingWikiWord' href='/nlab/show/Jordan+curve+theorem'>Jordan curve theorem</a></p>
446 </li>
447 </ul>
448
449 <p><strong>Analysis Theorems</strong></p>
450
451 <ul>
452 <li>
453 <p><a class='existingWikiWord' href='/nlab/show/Heine-Borel+theorem'>Heine-Borel theorem</a></p>
454 </li>
455
456 <li>
457 <p><a class='existingWikiWord' href='/nlab/show/intermediate+value+theorem'>intermediate value theorem</a></p>
458 </li>
459
460 <li>
461 <p><a class='existingWikiWord' href='/nlab/show/extreme+value+theorem'>extreme value theorem</a></p>
462 </li>
463 </ul>
464
465 <p><strong><a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological homotopy theory</a></strong></p>
466
467 <ul>
468 <li>
469 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
470 </li>
471
472 <li>
473 <p><a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/deformation+retract'>deformation retract</a></p>
474 </li>
475
476 <li>
477 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/covering+space'>covering space</a></p>
478 </li>
479
480 <li>
481 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
482 </li>
483
484 <li>
485 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
486 </li>
487
488 <li>
489 <p><a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a></p>
490 </li>
491
492 <li>
493 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead&#39;s theorem</a></p>
494 </li>
495
496 <li>
497 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
498 </li>
499
500 <li>
501 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
502 </li>
503
504 <li>
505 <p><a class='existingWikiWord' href='/nlab/show/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/Hurewicz+cofibration'>Hurewicz cofibration</a></p>
506 </li>
507
508 <li>
509 <p><a class='existingWikiWord' href='/nlab/show/topological+cofiber+sequence'>cofiber sequence</a></p>
510 </li>
511
512 <li>
513 <p><a class='existingWikiWord' href='/nlab/show/Str%C3%B8m+model+structure'>Strøm model category</a></p>
514 </li>
515
516 <li>
517 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p>
518 </li>
519 </ul>
520 </div>
521
522 <h4 id='homotopy_theory'>Homotopy theory</h4>
523
524 <div class='hide'>
525 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a></strong></p>
526
527 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/directed+homotopy+theory'>directed</a>…</p>
528
529 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/localic+homotopy+theory'>localic</a>, …</p>
530
531 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong></p>
532
533 <p><strong>Introductions</strong></p>
534
535 <ul>
536 <li>
537 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p>
538 </li>
539
540 <li>
541 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p>
542 </li>
543
544 <li>
545 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p>
546 </li>
547 </ul>
548
549 <p><strong>Definitions</strong></p>
550
551 <ul>
552 <li>
553 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/higher+homotopy'>higher homotopy</a></p>
554 </li>
555
556 <li>
557 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p>
558 </li>
559
560 <li>
561 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/spherical+object'>spherical object and Pi(A)-algebra</a></p>
562 </li>
563
564 <li>
565 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p>
566
567 <ul>
568 <li>
569 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></p>
570
571 <ul>
572 <li>
573 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p>
574 </li>
575
576 <li>
577 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/cofibration+category'>cofibration category</a></p>
578 </li>
579
580 <li>
581 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Waldhausen category</a></p>
582 </li>
583 </ul>
584 </li>
585
586 <li>
587 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></p>
588
589 <ul>
590 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a></li>
591 </ul>
592 </li>
593 </ul>
594 </li>
595
596 <li>
597 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
598
599 <ul>
600 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li>
601 </ul>
602 </li>
603 </ul>
604
605 <p><strong>Paths and cylinders</strong></p>
606
607 <ul>
608 <li>
609 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a></p>
610
611 <ul>
612 <li>
613 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a></p>
614 </li>
615
616 <li>
617 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a></p>
618 </li>
619 </ul>
620 </li>
621
622 <li>
623 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
624
625 <ul>
626 <li>
627 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path object</a></p>
628 </li>
629
630 <li>
631 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
632 </li>
633
634 <li>
635 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle'>universal bundle</a></p>
636 </li>
637 </ul>
638 </li>
639
640 <li>
641 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a></p>
642
643 <ul>
644 <li>
645 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometric+homotopies'>homotopy localization</a></p>
646 </li>
647
648 <li>
649 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+object'>infinitesimal interval object</a></p>
650 </li>
651 </ul>
652 </li>
653 </ul>
654
655 <p><strong>Homotopy groups</strong></p>
656
657 <ul>
658 <li>
659 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
660
661 <ul>
662 <li>
663 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a></p>
664
665 <ul>
666 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+topos'>fundamental group of a topos</a></li>
667 </ul>
668 </li>
669
670 <li>
671 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p>
672 </li>
673
674 <li>
675 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p>
676 </li>
677
678 <li>
679 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p>
680 </li>
681 </ul>
682 </li>
683
684 <li>
685 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p>
686
687 <ul>
688 <li>
689 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a></p>
690
691 <ul>
692 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path groupoid</a></li>
693 </ul>
694 </li>
695
696 <li>
697 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p>
698 </li>
699
700 <li>
701 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p>
702 </li>
703 </ul>
704 </li>
705
706 <li>
707 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p>
708
709 <ul>
710 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental category</a></li>
711 </ul>
712 </li>
713 </ul>
714
715 <p><strong>Basic facts</strong></p>
716
717 <ul>
718 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li>
719 </ul>
720
721 <p><strong>Theorems</strong></p>
722
723 <ul>
724 <li>
725 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
726 </li>
727
728 <li>
729 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
730 </li>
731
732 <li>
733 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p>
734 </li>
735
736 <li>
737 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p>
738 </li>
739
740 <li>
741 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
742 </li>
743
744 <li>
745 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead&#39;s theorem</a></p>
746 </li>
747
748 <li>
749 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewicz theorem</a></p>
750 </li>
751
752 <li>
753 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois theory</a></p>
754 </li>
755
756 <li>
757 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p>
758 </li>
759 </ul>
760 </div>
761 </div>
762 </div>
763
764 <h1 id='contents'>Contents</h1>
765 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#AsRightBaseChange'>As right base change along <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\ast \to \mathbf{B} S^1</annotation></semantics></math></a></li><li><a href='#ordinary_cohomology_of__on_cyclic_cohomology_of_'>Ordinary cohomology of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathcal{L}X \sslash S^1</annotation></semantics></math> on cyclic cohomology of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></a></li><li><a href='#rational_sullivan_model'>Rational Sullivan model</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div>
766 <h2 id='idea'>Idea</h2>
767
768 <p>Any <a class='existingWikiWord' href='/nlab/show/free+loop+space'>free loop space</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>\mathcal{L}X</annotation></semantics></math> has a canonical <a class='existingWikiWord' href='/nlab/show/action'>action</a> (<a class='existingWikiWord' href='/nlab/show/infinity-action'>infinity-action</a>) of the <a class='existingWikiWord' href='/nlab/show/circle+group'>circle group</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/homotopy+quotient'>homotopy quotient</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathcal{L}(X)/S^1</annotation></semantics></math> of this action might be called the <em>cyclic loop space</em> of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p>
769
770 <p>If <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X = Spec(A)</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/affine+variety'>affine variety</a> regarded in <a class='existingWikiWord' href='/nlab/show/derived+algebraic+geometry'>derived algebraic geometry</a>, then <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>Spec</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{O}(\mathcal{L}Spec(A))</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/Hochschild+cohomology'>Hochschild homology</a> of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>Spec</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{O}((\mathcal{L}Spec(A))/S^1)</annotation></semantics></math> the corresponding <a class='existingWikiWord' href='/nlab/show/cyclic+homology'>cyclic homology</a>, see the discussion at <em><a class='existingWikiWord' href='/nlab/show/Hochschild+cohomology'>Hochschild cohomology</a></em>.</p>
771
772 <p>If <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>X = Y//G</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/homotopy+quotient'>homotopy quotient</a> of a <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a> by a <a class='existingWikiWord' href='/nlab/show/topological+group'>topological group</a> action, regarded as a locally constant <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack, so that the <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>-action on <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{L}(X//G)</annotation></semantics></math> is an <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>B \mathbb{Z}</annotation></semantics></math>-action, then the restriction of the cyclic loop space to the constant loops <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℒ</mi> <mi>const</mi></msub><mi>Y</mi><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi><mo>→</mo><mi>ℒ</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{L}_{const}Y//G \to \mathcal{L}(Y//G)</annotation></semantics></math> has been called the <em>twisted loop space</em> in (<a href='#Witten88'>Witten 88</a>). This terminology has been widely adopted, for example in the context of the <a class='existingWikiWord' href='/nlab/show/transchromatic+character'>transchromatic character</a> map (<a href='#Stapleton11'>Stapleton 11</a>)</p>
773
774 <h2 id='properties'>Properties</h2>
775
776 <h3 id='AsRightBaseChange'>As right base change along <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\ast \to \mathbf{B} S^1</annotation></semantics></math></h3>
777
778 <p>The cyclic loop space <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathcal{L}X \sslash S^1</annotation></semantics></math> is equivalently the right <a class='existingWikiWord' href='/nlab/show/base+change'>base change</a>/<a class='existingWikiWord' href='/nlab/show/dependent+product'>dependent product</a> along the canonical point inclusion <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>→</mo><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\ast \to B S^1</annotation></semantics></math> (<a href='base+change#CyclicLoopSpace'>this prop.</a>) into the <a class='existingWikiWord' href='/nlab/show/delooping'>delooping</a> of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math> (the <a class='existingWikiWord' href='/nlab/show/classifying+space'>classifying space</a> of the <a class='existingWikiWord' href='/nlab/show/circle+group'>circle group</a> when realized in the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>homotopy theory of</a> <a class='existingWikiWord' href='/nlab/show/topological+space'>topological spaces</a>). See also at <em><a class='existingWikiWord' href='/nlab/show/double+dimensional+reduction'>double dimensional reduction</a></em> (<a href='#BMSS19'>BMSS 19, Sec. 2.2</a>, following <a href='#FSS18'>FSS 18, Sec. 3</a>).</p>
779
780 <h3 id='ordinary_cohomology_of__on_cyclic_cohomology_of_'>Ordinary cohomology of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathcal{L}X \sslash S^1</annotation></semantics></math> on cyclic cohomology of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></h3>
781
782 <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/simply+connected+space'>simply connected</a> <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a>.</p>
783
784 <p>The <a class='existingWikiWord' href='/nlab/show/ordinary+cohomology'>ordinary cohomology</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding='application/x-tex'>H^\bullet</annotation></semantics></math> of its <a class='existingWikiWord' href='/nlab/show/free+loop+space'>free loop space</a> is the <a class='existingWikiWord' href='/nlab/show/Hochschild+cohomology'>Hochschild homology</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>HH</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>HH_\bullet</annotation></semantics></math> of its <a class='existingWikiWord' href='/nlab/show/singular+cohomology'>singular chains</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet(X)</annotation></semantics></math>:</p>
785 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>HH</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
786 H^\bullet(\mathcal{L}X)
787 \simeq
788 HH_\bullet( C^\bullet(X) )
789 \,.
790
791 </annotation></semantics></math></div>
792 <p>Moreover the <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathcal{L}X \sslash S^1</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/cyclic+homology'>cyclic homology</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>HC</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>HC_\bullet</annotation></semantics></math> of the singular chains:</p>
793 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>HC</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
794 H^\bullet(\mathcal{L}X \sslash S^1)
795 \simeq
796 HC_\bullet( C^\bullet(X) )
797
798 </annotation></semantics></math></div>
799 <p>(<a href='#Jones87'>Jones 87, Thm. A</a>, review in <a href='#Loday92'>Loday 92, Cor. 7.3.14</a>, <a href='#Loday11'>Loday 11, Sec. 4</a>)</p>
800
801 <p>If the <a class='existingWikiWord' href='/nlab/show/coefficient'>coefficients</a> are <a class='existingWikiWord' href='/nlab/show/rational+number'>rational</a>, and <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is of <a class='existingWikiWord' href='/nlab/show/finite+type'>finite type</a> then this may be computed by the <em><a class='existingWikiWord' href='/nlab/show/Sullivan+model+of+loop+space'>Sullivan model for free loop spaces</a></em>, see there the section on <em><a href='Sullivan+model+of+free+loop+space#RelationToHochschildHomology'>Relation to Hochschild homology</a></em>.</p>
802
803 <p>In the special case that the <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> carries the structure of a <a class='existingWikiWord' href='/nlab/show/smooth+manifold'>smooth manifold</a>, then the singular cochains on <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> are equivalent to the <a class='existingWikiWord' href='/nlab/show/differential+graded-commutative+algebra'>dgc-algebra</a> of <a class='existingWikiWord' href='/nlab/show/differential+form'>differential forms</a> (the <a class='existingWikiWord' href='/nlab/show/de+Rham+complex'>de Rham algebra</a>) and hence in this case the statement becomes that</p>
804 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>HH</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
805 H^\bullet(\mathcal{L}X)
806 \simeq
807 HH_\bullet( \Omega^\bullet(X) )
808 \,.
809
810 </annotation></semantics></math></div><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>ℒ</mi><mi>X</mi><mo>⫽</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>HC</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
811 H^\bullet(\mathcal{L}X \sslash S^1)
812 \simeq
813 HC_\bullet( \Omega^\bullet(X) )
814 \,.
815
816 </annotation></semantics></math></div>
817 <p>This is known as <em><a class='existingWikiWord' href='/nlab/show/Jones%27+theorem'>Jones&#39; theorem</a></em> (<a href='#Jones87'>Jones 87</a>)</p>
818
819 <p>An <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>infinity-category theoretic</a> proof of this fact is indicated at <em><a href='Hochschild+cohomology#JonesTheorem'>Hochschild cohomology – Jones’ theorem</a></em>.</p>
820
821 <h3 id='rational_sullivan_model'>Rational Sullivan model</h3>
822
823 <p>See at <em><a class='existingWikiWord' href='/nlab/show/Sullivan+model+of+loop+space'>Sullivan model for free loop space</a></em></p>
824
825 <h2 id='related_concepts'>Related concepts</h2>
826
827 <ul>
828 <li>
829 <p><a class='existingWikiWord' href='/nlab/show/double+dimensional+reduction'>double dimensional reduction</a></p>
830 </li>
831
832 <li>
833 <p><a class='existingWikiWord' href='/nlab/show/cyclic+loop+stack'>cyclic loop stack</a></p>
834 </li>
835
836 <li>
837 <p><a class='existingWikiWord' href='/nlab/show/free+loop+space'>free loop space</a>, <a class='existingWikiWord' href='/nlab/show/free+loop+orbifold'>free loop stack</a></p>
838 </li>
839 </ul>
840
841 <h2 id='references'>References</h2>
842
843 <p>The notion of the cyclic loop space of a topological space appears as:</p>
844
845 <ul>
846 <li id='Jones87'>
847 <p><a class='existingWikiWord' href='/nlab/show/John+David+Stuart+Jones'>John D.S. Jones</a>, <em>Cyclic homology and equivariant homology</em>, Invent. Math. <strong>87</strong>, 403-423 (1987) (<a href='https://math.berkeley.edu/~nadler/jones.pdf'>pdf</a>, <a href='https://doi.org/10.1007/BF01389424'>doi:10.1007/BF01389424</a>)</p>
848 </li>
849
850 <li>
851 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>The cyclic groups and the free loop space</em>, Commentarii Mathematici Helvetici <strong>62</strong> (1987) 423–449 (<a href='https://doi.org/10.1007/BF02564455'>doi:10.1007/BF02564455</a>, <a href='https://eudml.org/doc/140092'>dml:140092</a>)</p>
852 </li>
853
854 <li id='Witten88'>
855 <p><a class='existingWikiWord' href='/nlab/show/Edward+Witten'>Edward Witten</a>, <em>The index of the Dirac operator in loop space</em>. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), volume 1326 of Lecture Notes in Math., pages 161–181. Springer, Berlin, 1988 (<a href='https://doi.org/10.1007/BFb0078045'>doi:10.1007/BFb0078045</a>)</p>
856 </li>
857
858 <li id='Loday92'>
859 <p><a class='existingWikiWord' href='/nlab/show/Jean-Louis+Loday'>Jean-Louis Loday</a>, <em>Cyclic Spaces and <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>-Equivariant Homology</em> (<a href='https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7'>doi:10.1007/978-3-662-21739-9_7</a>)</p>
860
861 <p>Chapter 7 in: <em>Cyclic Homology</em>, Grundlehren <strong>301</strong>, Springer 1992 (<a href='https://link.springer.com/book/10.1007/978-3-662-21739-9'>doi:10.1007/978-3-662-21739-9</a>)</p>
862 </li>
863
864 <li id='Loday11'>
865 <p><a class='existingWikiWord' href='/nlab/show/Jean-Louis+Loday'>Jean-Louis Loday</a>, Section 4 of: <em>Free loop space and homology</em>, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): <em>Free Loop Spaces in Geometry and Topology</em>, IRMA Lectures in Mathematics and Theoretical Physics <strong>24</strong>, EMS 2015 (<a href='https://arxiv.org/abs/1110.0405'>arXiv:1110.0405</a>, <a href='https://bookstore.ams.org/emsilmtp-24/'>ISBN:978-3-03719-153-8</a>)</p>
866 </li>
867
868 <li id='Stapleton11'>
869 <p><a class='existingWikiWord' href='/nlab/show/Nathaniel+Stapleton'>Nathaniel Stapleton</a>, <em>Transchromatic generalized character maps</em>, Algebr. Geom. Topol. 13 (2013) 171-203 (<a href='https://arxiv.org/abs/1110.3346'>arXiv:1110.3346</a>)</p>
870 </li>
871 </ul>
872
873 <p>Specifically on cyclic loop spaces of <a class='existingWikiWord' href='/nlab/show/sphere'>n-spheres</a>:</p>
874
875 <ul>
876 <li><a class='existingWikiWord' href='/nlab/show/Nancy+Hingston'>Nancy Hingston</a>, <em>An Equivariant Model for the Free Loop Space of <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4bd1bbcadb692754f6cd72dfe8e06a9b_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>S^N</annotation></semantics></math></em>, American Journal of Mathematics <strong>114</strong> 1 (1992) 139-155 (<a href='https://doi.org/10.2307/2374740'>doi:10.2307/2374740</a>, <a href='https://www.jstor.org/stable/2374740'>jstor:2374740</a>)</li>
877 </ul>
878
879 <p>See also:</p>
880
881 <ul>
882 <li><a class='existingWikiWord' href='/nlab/show/Urs+Frauenfelder'>Urs Frauenfelder</a>, <em>Dihedral homology and the moon</em>, J. Fixed Point Theory Appl. <strong>14</strong> (2013) 55–69 (<a href='https://arxiv.org/abs/1204.4549'>arXiv:1204.4549</a>, <a href='https://doi.org/10.1007/s11784-013-0146-z'>doi:10.1007/s11784-013-0146-z</a>)</li>
883 </ul>
884
885 <p>A version of the cyclic loop space of <a class='existingWikiWord' href='/nlab/show/orbifold'>orbifolds</a>, or at least its restriction to constant loops, namely <a class='existingWikiWord' href='/nlab/show/Huan%27s+inertia+orbifold'>Huan&#39;s inertia orbifold</a>, is discussed in the context of <a class='existingWikiWord' href='/nlab/show/equivariant+elliptic+cohomology'>equivariant elliptic cohomology</a> via <a class='existingWikiWord' href='/nlab/show/Tate+K-theory'>Tate K-theory</a> in:</p>
886
887 <ul>
888 <li id='Huan18'><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>Zhen Huan</a>, Def. 2.14 of: <em>Quasi-Elliptic Cohomology I</em>, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (<a href='https://arxiv.org/abs/1805.06305'>arXiv:1805.06305</a>, <a href='https://doi.org/10.1016/j.aim.2018.08.007'>doi:10.1016/j.aim.2018.08.007</a>)</li>
889 </ul>
890
891 <p>following</p>
892
893 <ul>
894 <li><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>Zhen Huan</a>, Section 2.1.2 of: <em>Quasi-elliptic cohomology</em>, 2017 (<a href='http://hdl.handle.net/2142/97268'>hdl</a>)</li>
895 </ul>
896
897 <p>and recalled/expanded on in several followup articles, such as in</p>
898
899 <ul>
900 <li><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>Zhen Huan</a>, Section 2 of <em>Quasi-theories</em> (<a href='https://arxiv.org/abs/1809.06651'>arXiv:1809.06651</a>)</li>
901 </ul>
902
903 <p>The above formulation of cyclic loop spaces, in the generality of <a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stacks</a>, as right <a class='existingWikiWord' href='/nlab/show/base+change'>base change</a> to the <a class='existingWikiWord' href='/nlab/show/delooping'>delooping</a> of the <a class='existingWikiWord' href='/nlab/show/circle+group'>circle group</a>, and its relation to <a class='existingWikiWord' href='/nlab/show/double+dimensional+reduction'>double dimensional reduction</a> in <a class='existingWikiWord' href='/nlab/show/brane'>brane</a>-physics, is due to:</p>
904
905 <ul>
906 <li id='BMSS19'><a class='existingWikiWord' href='/nlab/show/Vincent+Braunack-Mayer'>Vincent Braunack-Mayer</a>, <a class='existingWikiWord' href='/nlab/show/Hisham+Sati'>Hisham Sati</a>, <a class='existingWikiWord' href='/nlab/show/Urs+Schreiber'>Urs Schreiber</a>: Section 2.2 of <em><a class='existingWikiWord' href='/schreiber/show/Gauge+enhancement+of+Super+M-Branes' title='schreiber'>Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory</a></em>, Communications in Mathematical Physics, <strong>371</strong> 197 (2019) (<a href='https://doi.org/10.1007/s00220-019-03441-4'>doi:10.1007/s00220-019-03441-4</a>, <a href='https://arxiv.org/abs/1806.01115'>arXiv:1806.01115</a>)</li>
907 </ul>
908
909 <p>following the analogous discussion in <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational homotopy theory</a> in</p>
910
911 <ul>
912 <li id='FSS18'><a class='existingWikiWord' href='/nlab/show/Domenico+Fiorenza'>Domenico Fiorenza</a>, <a class='existingWikiWord' href='/nlab/show/Hisham+Sati'>Hisham Sati</a>, <a class='existingWikiWord' href='/nlab/show/Urs+Schreiber'>Urs Schreiber</a>, Section 3 of: <em><a class='existingWikiWord' href='/schreiber/show/T-Duality+from+super+Lie+n-algebra+cocycles+for+super+p-branes' title='schreiber'>T-Duality from super Lie $n$-algebra cocycles for super $p$-branes</a></em>, <a href='http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/'>ATMP Volume 22 (2018) Number 5</a>, <a href='http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3'>doi:10.4310/ATMP.2018.v22.n5.a3</a>, <a href='https://arxiv.org/abs/1611.06536'>arXiv:1611.06536</a>)</li>
913 </ul>
914
915 <p>with exposition in</p>
916
917 <ul>
918 <li><a class='existingWikiWord' href='/nlab/show/Urs+Schreiber'>Urs Schreiber</a>, <a href='https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction'>Section 4</a> of: <em><a class='existingWikiWord' href='/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes' title='schreiber'>Super Lie n-algebra of Super p-branes</a></em> (2016)</li>
919 </ul>
920
921 <p>
922 </p>
923
924 <p>
925
926 </p> </div>
927 </content>
928 </entry>
929 <entry>
930 <title type="html">Nancy Hingston</title>
931 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Nancy+Hingston"/>
932 <updated>2021-07-02T09:15:40Z</updated>
933 <published>2021-07-02T09:14:11Z</published>
934 <id>tag:ncatlab.org,2021-07-02:nLab,Nancy+Hingston</id>
935 <author>
936 <name>Urs Schreiber</name>
937 </author>
938 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Nancy+Hingston">
939 <div xmlns="http://www.w3.org/1999/xhtml">
940 <ul>
941 <li>
942 <p><a href='https://en.wikipedia.org/wiki/Nancy_Hingston'>Wikipedia entry</a></p>
943 </li>
944
945 <li>
946 <p><a href='https://science.tcnj.edu/school-information/women-in-science/dr-nancy-hingston/'>Institute page</a></p>
947 </li>
948
949 <li>
950 <p><a href='https://science.tcnj.edu/school-information/women-in-science/dr-nancy-hingston/'>Mathematics Genealogy page</a></p>
951 </li>
952 </ul>
953
954 <h2 id='selected_writings'>Selected writings</h2>
955
956 <p>On the <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cyclic loop spaces</a> of <a class='existingWikiWord' href='/nlab/show/sphere'>n-spheres</a>:</p>
957
958 <ul>
959 <li><a class='existingWikiWord' href='/nlab/show/Nancy+Hingston'>Nancy Hingston</a>, <em>An Equivariant Model for the Free Loop Space of <math class='maruku-mathml' display='inline' id='mathml_6597653e93efa254ef9530d15b5b7be84786eb66_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>S^N</annotation></semantics></math></em>, American Journal of Mathematics <strong>114</strong> 1 (1992) 139-155 (<a href='https://doi.org/10.2307/2374740'>doi:10.2307/2374740</a>, <a href='https://www.jstor.org/stable/2374740'>jstor:2374740</a>)</li>
960 </ul>
961
962 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p> </div>
963 </content>
964 </entry>
965 <entry>
966 <title type="html">Ralph Cohen</title>
967 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Ralph+Cohen"/>
968 <updated>2021-07-02T08:58:22Z</updated>
969 <published>2010-11-26T21:46:21Z</published>
970 <id>tag:ncatlab.org,2010-11-26:nLab,Ralph+Cohen</id>
971 <author>
972 <name>Urs Schreiber</name>
973 </author>
974 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Ralph+Cohen">
975 <div xmlns="http://www.w3.org/1999/xhtml">
976 <ul>
977 <li><a href='http://math.stanford.edu/~ralph/'>website</a></li>
978 </ul>
979
980 <h2 id='selected_writings'>Selected writings</h2>
981
982 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cyclic loop spaces</a>:</p>
983
984 <ul>
985 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>The cyclic groups and the free loop space</em>, Commentarii Mathematici Helvetici <strong>62</strong> (1987) 423–449 (<a href='https://doi.org/10.1007/BF02564455'>doi:10.1007/BF02564455</a>, <a href='https://eudml.org/doc/140092'>dml:140092</a>)</li>
986 </ul>
987
988 <p>On <a class='existingWikiWord' href='/nlab/show/string+topology'>string topology</a>:</p>
989
990 <ul>
991 <li>
992 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, John R. Klein, <a class='existingWikiWord' href='/nlab/show/Dennis+Sullivan'>Dennis Sullivan</a>, <em>The homotopy invariance of the string topology loop product and string bracket</em>, J. of Topology 2008 <strong>1</strong>(2):391-408; <a href='http://dx.doi.org/10.1112/jtopol/jtn001'>doi</a></p>
993 </li>
994
995 <li>
996 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>Homotopy and geometric perspectives on string topology</em> (<a href='http://math.stanford.edu/~ralph/skyesummary.pdf'>pdf</a>)</p>
997 </li>
998
999 <li id='CohenJones'>
1000 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <a class='existingWikiWord' href='/nlab/show/John+David+Stuart+Jones'>John David Stuart Jones</a>, <em>A homotopy theoretic realization of string topology</em> , Math. Ann. 324 (2002), no. 4, (<a href='http://arxiv.org/abs/math/0107187'>arXiv:0107187</a>)</p>
1001 </li>
1002
1003 <li id='CohenGodin03'>
1004 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <a class='existingWikiWord' href='/nlab/show/Veronique+Godin'>Veronique Godin</a>, <em><a class='existingWikiWord' href='/nlab/show/A+Polarized+View+of+String+Topology'>A Polarized View of String Topology</a></em> (<a href='http://arxiv.org/abs/math/0303003'>arXiv:math/0303003</a>)</p>
1005 </li>
1006
1007 <li>
1008 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <a class='existingWikiWord' href='/nlab/show/Alexander+Voronov'>Alexander Voronov</a>, <em>Notes on string topology</em>, in: <a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <a class='existingWikiWord' href='/nlab/show/Kathryn+Hess'>Kathryn Hess</a>, <a class='existingWikiWord' href='/nlab/show/Alexander+Voronov'>Alexander Voronov</a>, <em>String topology and cyclic homology</em>, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 (<a href='http://arxiv.org/abs/math/0503625'>math.GT/05036259</a>, <a href='https://doi.org/10.1007/3-7643-7388-1'>doi:10.1007/3-7643-7388-1</a>, <a href='http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7'>pdf</a>)</p>
1009 </li>
1010
1011 <li>
1012 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <a class='existingWikiWord' href='/nlab/show/John+David+Stuart+Jones'>John Jones</a>, <em>Gauge theory and string topology</em> (<a href='http://arxiv.org/abs/1304.0613'>arXiv:1304.0613</a>)</p>
1013 </li>
1014 </ul>
1015
1016 <p>On <a class='existingWikiWord' href='/nlab/show/moduli+space+of+monopoles'>moduli spaces of monopoles</a> related to <a class='existingWikiWord' href='/nlab/show/braid+group'>braid groups</a>:</p>
1017
1018 <ul>
1019 <li>
1020 <p><a class='existingWikiWord' href='/nlab/show/Fred+Cohen'>Fred Cohen</a>, <a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, B. M. Mann, R. J. Milgram, <em>The topology of rational functions and divisors of surfaces</em>, Acta Math (1991) 166: 163 (<a href='https://doi.org/10.1007/BF02398886'>doi:10.1007/BF02398886</a>)</p>
1021 </li>
1022
1023 <li>
1024 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, John D. S. Jones <em>Monopoles, braid groups, and the Dirac operator</em>, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 (<a href='https://projecteuclid.org/euclid.cmp/1104254240'>euclid:cmp/1104254240</a>)</p>
1025 </li>
1026 </ul>
1027
1028 <p>and more generally on <a class='existingWikiWord' href='/nlab/show/moduli+space'>moduli spaces</a>:</p>
1029
1030 <ul>
1031 <li><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>Stability phenomena in the topology of moduli spaces</em> (<a href='https://arxiv.org/abs/0908.1938'>arXiv:0908.1938</a>)</li>
1032 </ul>
1033
1034 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p> </div>
1035 </content>
1036 </entry>
1037 <entry>
1038 <title type="html">Gunnar Carlsson</title>
1039 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Gunnar+Carlsson"/>
1040 <updated>2021-07-02T08:57:51Z</updated>
1041 <published>2014-04-13T09:19:17Z</published>
1042 <id>tag:ncatlab.org,2014-04-13:nLab,Gunnar+Carlsson</id>
1043 <author>
1044 <name>Urs Schreiber</name>
1045 </author>
1046 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Gunnar+Carlsson">
1047 <div xmlns="http://www.w3.org/1999/xhtml">
1048 <ul>
1049 <li><a href='http://math.stanford.edu/~gunnar/'>webpage</a></li>
1050 </ul>
1051
1052 <h2 id='selected_writings'>Selected writings</h2>
1053
1054 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cyclic loop spaces</a>:</p>
1055
1056 <ul>
1057 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>The cyclic groups and the free loop space</em>, Commentarii Mathematici Helvetici <strong>62</strong> (1987) 423–449 (<a href='https://doi.org/10.1007/BF02564455'>doi:10.1007/BF02564455</a>, <a href='https://eudml.org/doc/140092'>dml:140092</a>)</li>
1058 </ul>
1059
1060 <p>On <a class='existingWikiWord' href='/nlab/show/topological+data+analysis'>topological data analysis</a>:</p>
1061
1062 <ul>
1063 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <em>Topology and data</em>, Bull. Amer. Math. Soc. 46 (2009), 255-308 (<a href='https://doi.org/10.1090/S0273-0979-09-01249-X'>doi:10.1090/S0273-0979-09-01249-X</a>)</li>
1064 </ul>
1065
1066 <p>On <a class='existingWikiWord' href='/nlab/show/persistent+homology'>persistent homology</a>:</p>
1067
1068 <ul>
1069 <li>
1070 <p>A. Zomorodian, <a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <em>Computing persistent homology</em>, Discrete Comput. Geom. <strong>33</strong>, 249–274 (2005)</p>
1071 </li>
1072
1073 <li>
1074 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, V. de Silva, <em>Zigzag persistence</em>, <a href='http://arxiv.org/abs/0812.0197'>arXiv:0812.0197</a></p>
1075 </li>
1076
1077 <li>
1078 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar Carlsson</a>, <em>Persistent Homology and Applied Homotopy Theory</em>, in: <a class='existingWikiWord' href='/nlab/show/Handbook+of+Homotopy+Theory'>Handbook of Homotopy Theory</a>, CRC Press, 2019 (<a href='https://arxiv.org/abs/2004.00738'>arXiv:2004.00738</a>)</p>
1079 </li>
1080 </ul>
1081
1082 <h2 id='related_lab_entries'>Related <math class='maruku-mathml' display='inline' id='mathml_2134ce935eca0d191efd191c47bb046d957cb8f1_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>Lab entries</h2>
1083
1084 <ul>
1085 <li>
1086 <p><a class='existingWikiWord' href='/nlab/show/equivariant+stable+homotopy+theory'>equivariant stable homotopy theory</a></p>
1087 </li>
1088
1089 <li>
1090 <p><a class='existingWikiWord' href='/nlab/show/Segal-Carlsson+completion+theorem'>Segal conjecture</a>, <a class='existingWikiWord' href='/nlab/show/Sullivan+conjecture'>Sullivan conjecture</a></p>
1091
1092 <p><a class='existingWikiWord' href='/nlab/show/Burnside+ring'>Burnside ring</a>, <a class='existingWikiWord' href='/nlab/show/stable+cohomotopy'>stable cohomotopy</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+stable+cohomotopy'>equivariant stable cohomotopy</a></p>
1093 </li>
1094
1095 <li>
1096 <p><a class='existingWikiWord' href='/nlab/show/persistent+homology'>persistent homology</a></p>
1097 </li>
1098 </ul>
1099
1100 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p>
1101
1102 <p>
1103 </p> </div>
1104 </content>
1105 </entry>
1106 <entry>
1107 <title type="html">Doug Ravenel</title>
1108 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Doug+Ravenel"/>
1109 <updated>2021-07-02T08:31:03Z</updated>
1110 <published>2012-08-14T17:36:28Z</published>
1111 <id>tag:ncatlab.org,2012-08-14:nLab,Doug+Ravenel</id>
1112 <author>
1113 <name>Urs Schreiber</name>
1114 </author>
1115 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Doug+Ravenel">
1116 <div xmlns="http://www.w3.org/1999/xhtml">
1117 <ul>
1118 <li>
1119 <p><a href='http://www.math.rochester.edu/people/faculty/doug/'>webpage</a></p>
1120 </li>
1121
1122 <li>
1123 <p><a class='existingWikiWord' href='/nlab/show/Michael+Hopkins'>Michael Hopkins</a>, <em>The mathematical work of Douglas C. Ravenel</em>, Homology Homotopy Appl. Volume 10, Number 3 (2008), 1-13 (<a href='https://projecteuclid.org/euclid.hha/1251832464'>euclid:hha/1251832464</a>)</p>
1124 </li>
1125 </ul>
1126
1127 <h2 id='selected_writings'>Selected writings</h2>
1128
1129 <p>On the <a class='existingWikiWord' href='/nlab/show/Hopf+algebra'>Hopf ring</a> of <a class='existingWikiWord' href='/nlab/show/MU'>MU</a>:</p>
1130
1131 <ul>
1132 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <a class='existingWikiWord' href='/nlab/show/W.+Stephen+Wilson'>W. Stephen Wilson</a>, <em>The Hopf ring for complex cobordism</em>, Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (<a href='https://doi.org/10.1016/0022-4049(77)90070-6'>doi:10.1016/0022-4049(77)90070-6</a>, <a href='https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink'>euclid</a>, <a href='https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf'>pdf</a>)</li>
1133 </ul>
1134
1135 <p>On the <a class='existingWikiWord' href='/nlab/show/Adams%E2%80%93Novikov+spectral+sequence'>Adams-Novikov spectral sequence</a>:</p>
1136
1137 <ul>
1138 <li id='Ravenel78'><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <em>A Novice’s guide to the Adams-Novikov spectral sequence</em>, in: <a class='existingWikiWord' href='/nlab/show/Michael+Barratt'>Michael Barratt</a>, <a class='existingWikiWord' href='/nlab/show/Mark+Mahowald'>Mark Mahowald</a> (eds.) <em>Geometric Applications of Homotopy Theory II</em>. Lecture Notes in Mathematics, vol 658, Springer 1978 (<a href='https://doi.org/10.1007/BFb0068728'>doi:10.1007/BFb0068728</a>, <a href='https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf'>pdf</a>)</li>
1139 </ul>
1140
1141 <p>On <a class='existingWikiWord' href='/nlab/show/chromatic+homotopy+theory'>chromatic homotopy theory</a> and introducing <a class='existingWikiWord' href='/nlab/show/Ravenel%27s+spectrum'>Ravenel&#39;s spectra</a> and <a class='existingWikiWord' href='/nlab/show/Ravenel%27s+conjectures'>Ravenel&#39;s conjectures</a>:</p>
1142
1143 <ul>
1144 <li id='Ravenel84'><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <em>Localization with Respect to Certain Periodic Homology Theories</em>, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (<a href='https://doi.org/10.2307/2374308'>doi:10.2307/2374308</a>, <a href='https://www.jstor.org/stable/2374308'>jstor:2374308</a>)</li>
1145 </ul>
1146
1147 <p>On <a class='existingWikiWord' href='/nlab/show/homotopy+groups+of+spheres'>stable homotopy groups of spheres</a> and <a class='existingWikiWord' href='/nlab/show/chromatic+homotopy+theory'>chromatic homotopy theory</a>:</p>
1148
1149 <ul>
1150 <li id='MahowaldRavenel87'><a class='existingWikiWord' href='/nlab/show/Mark+Mahowald'>Mark Mahowald</a>, <a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Doug Ravenel</a>, <em>Towards a Global Understanding of the Homotopy Groups of Spheres</em>, in: <a class='existingWikiWord' href='/nlab/show/Samuel+Gitler'>Samuel Gitler</a> (ed.): <em>The Lefschetz Centennial Conference: Proceedings on Algebraic Topology II</em>, Contemporary Mathematics volume 58, AMS 1987 (<a href='https://bookstore.ams.org/conm-58-2'>ISBN:978-0-8218-5063-3</a>, <a href='http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/MahowaldRavenelHomotopyGroupsOfSpheres.pdf' title='pdf'>pdf</a>)</li>
1151 </ul>
1152
1153 <p>On <a class='existingWikiWord' href='/nlab/show/elliptic+genus'>elliptic genera</a>:</p>
1154
1155 <ul>
1156 <li id='LandweberRavenelStong93'><a class='existingWikiWord' href='/nlab/show/Peter+Landweber'>Peter Landweber</a>, <a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <a class='existingWikiWord' href='/nlab/show/Robert+Stong'>Robert Stong</a>, <em>Periodic cohomology theories defined by elliptic curves</em>, in <a class='existingWikiWord' href='/nlab/show/Haynes+Miller'>Haynes Miller</a> et al (eds.), <em>The Cech centennial: A conference on homotopy theory</em>, June 1993, AMS (1995) (<a href='http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf'>pdf</a>)</li>
1157 </ul>
1158
1159 <p>On <a class='existingWikiWord' href='/nlab/show/chromatic+homotopy+theory'>chromatic homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/MU'>complex cobordism cohomology</a> and <a class='existingWikiWord' href='/nlab/show/homotopy+groups+of+spheres'>stable homotopy groups of spheres</a>,:</p>
1160
1161 <ul>
1162 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Doug Ravenel</a>, <em><a class='existingWikiWord' href='/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres'>Complex cobordism and stable homotopy groups of spheres</a></em>, Academic Press Orland (1986) reprinted as: AMS Chelsea Publishing, Volume 347, 2004 (<a href='https://bookstore.ams.org/chel-347-h'>ISBN:978-0-8218-2967-7</a>, <a href='http://www.math.rochester.edu/people/faculty/doug/mu.html'>webpage</a>, <a href='https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf'>pdf</a>)</li>
1163 </ul>
1164
1165 <p>On <a class='existingWikiWord' href='/nlab/show/iterated+loop+space'>iterated loop spaces</a> of <a class='existingWikiWord' href='/nlab/show/sphere'>spheres</a> and <a class='existingWikiWord' href='/nlab/show/stable+splitting+of+mapping+spaces'>stable splitting of mapping spaces</a>:</p>
1166
1167 <ul>
1168 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <em>What we still don’t understand about loop spaces of spheres</em>, Contemporary Mathematics 1998 (<a href='https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Ravenel_LoopSpacesOfSpheres.pdf' title='pdf'>pdf</a>)</li>
1169 </ul>
1170
1171 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p>
1172
1173 <p>
1174 </p> </div>
1175 </content>
1176 </entry>
1177 <entry>
1178 <title type="html">stable splitting of mapping spaces</title>
1179 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/stable+splitting+of+mapping+spaces"/>
1180 <updated>2021-07-02T08:30:06Z</updated>
1181 <published>2018-10-28T12:06:36Z</published>
1182 <id>tag:ncatlab.org,2018-10-28:nLab,stable+splitting+of+mapping+spaces</id>
1183 <author>
1184 <name>Urs Schreiber</name>
1185 </author>
1186 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/stable+splitting+of+mapping+spaces">
1187 <div xmlns="http://www.w3.org/1999/xhtml">
1188 <div class='rightHandSide'>
1189 <div class='toc clickDown' tabindex='0'>
1190 <h3 id='context'>Context</h3>
1191
1192 <h4 id='stable_homotopy_theory'>Stable Homotopy theory</h4>
1193
1194 <div class='hide'>
1195 <p><strong><a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable homotopy theory</a></strong></p>
1196
1197 <ul>
1198 <li><a class='existingWikiWord' href='/nlab/show/homological+algebra'>homological algebra</a>, <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a></li>
1199 </ul>
1200
1201 <p><em><a class='existingWikiWord' href='/nlab/show/Introduction+to+Stable+Homotopy+Theory'>Introduction</a></em></p>
1202
1203 <h1 id='ingredients'>Ingredients</h1>
1204
1205 <ul>
1206 <li><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></li>
1207 </ul>
1208
1209 <h1 id='contents'>Contents</h1>
1210
1211 <ul>
1212 <li>
1213 <p><a class='existingWikiWord' href='/nlab/show/loop+space+object'>loop space object</a></p>
1214 </li>
1215
1216 <li>
1217 <p><a class='existingWikiWord' href='/nlab/show/suspension+object'>suspension object</a></p>
1218 </li>
1219
1220 <li>
1221 <p><a class='existingWikiWord' href='/nlab/show/looping'>looping and delooping</a></p>
1222 </li>
1223
1224 <li>
1225 <p><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p>
1226
1227 <ul>
1228 <li>
1229 <p><a class='existingWikiWord' href='/nlab/show/stabilization'>stabilization</a></p>
1230
1231 <ul>
1232 <li><a class='existingWikiWord' href='/nlab/show/spectrum+object'>spectrum object</a></li>
1233 </ul>
1234 </li>
1235
1236 <li>
1237 <p><a class='existingWikiWord' href='/nlab/show/stable+derivator'>stable derivator</a></p>
1238 </li>
1239
1240 <li>
1241 <p><a class='existingWikiWord' href='/nlab/show/triangulated+category'>triangulated category</a></p>
1242 </li>
1243 </ul>
1244 </li>
1245
1246 <li>
1247 <p><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%2C1%29-category+of+spectra'>stable (∞,1)-category of spectra</a></p>
1248
1249 <ul>
1250 <li>
1251 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></p>
1252 </li>
1253
1254 <li>
1255 <p><a class='existingWikiWord' href='/nlab/show/stable+homotopy+category'>stable homotopy category</a></p>
1256 </li>
1257 </ul>
1258 </li>
1259
1260 <li>
1261 <p><a class='existingWikiWord' href='/nlab/show/smash+product+of+spectra'>smash product of spectra</a></p>
1262
1263 <ul>
1264 <li>
1265 <p><a class='existingWikiWord' href='/nlab/show/symmetric+smash+product+of+spectra'>symmetric smash product of spectra</a></p>
1266 </li>
1267
1268 <li>
1269 <p><a class='existingWikiWord' href='/nlab/show/Spanier-Whitehead+duality'>Spanier-Whitehead duality</a></p>
1270 </li>
1271
1272 <li>
1273 <p><a class='existingWikiWord' href='/nlab/show/A-infinity-ring'>A-∞ ring</a></p>
1274 </li>
1275
1276 <li>
1277 <p><a class='existingWikiWord' href='/nlab/show/E-infinity-ring'>E-∞ ring</a></p>
1278 </li>
1279 </ul>
1280 </li>
1281 </ul>
1282 <div>
1283 <p>
1284 <a href='/nlab/edit/stable+homotopy+theory+-+contents'>Edit this sidebar</a>
1285 </p>
1286 </div></div>
1287
1288 <h4 id='goodwillie_calculus'>Goodwillie calculus</h4>
1289
1290 <div class='hide'>
1291 <p><strong><a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie calculus</a></strong> – approximation of <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theories</a> by <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable homotopy theories</a></p>
1292
1293 <ul>
1294 <li>
1295 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie-differentiable+%28infinity%2C1%29-category'>Goodwillie-differentiable (∞,1)-category</a></p>
1296 </li>
1297
1298 <li>
1299 <p><a class='existingWikiWord' href='/nlab/show/excisive+%28%E2%88%9E%2C1%29-functor'>excisive (∞,1)-functor</a></p>
1300
1301 <ul>
1302 <li>
1303 <p><a class='existingWikiWord' href='/nlab/show/spectrum+object'>spectrum object</a>, <a class='existingWikiWord' href='/nlab/show/parametrized+spectrum'>parameterized spectrum</a>,</p>
1304 </li>
1305
1306 <li>
1307 <p><a class='existingWikiWord' href='/nlab/show/tangent+%28infinity%2C1%29-category'>tangent (∞,1)-category</a>, <a class='existingWikiWord' href='/nlab/show/tangent+%28infinity%2C1%29-category'>tangent (∞,1)-topos</a></p>
1308 </li>
1309 </ul>
1310 </li>
1311
1312 <li>
1313 <p><a class='existingWikiWord' href='/nlab/show/n-excisive+%28%E2%88%9E%2C1%29-functor'>n-excisive (∞,1)-functor</a></p>
1314
1315 <ul>
1316 <li>
1317 <p><a class='existingWikiWord' href='/nlab/show/jet+%28infinity%2C1%29-category'>jet (∞,1)-category</a></p>
1318 </li>
1319
1320 <li>
1321 <p><a class='existingWikiWord' href='/nlab/show/polynomial+%28%E2%88%9E%2C1%29-functor'>polynomial (∞,1)-functor</a>, <a class='existingWikiWord' href='/nlab/show/n-reduced+%28%E2%88%9E%2C1%29-functor'>n-reduced (∞,1)-functor</a>, <a class='existingWikiWord' href='/nlab/show/n-homogeneous+%28%E2%88%9E%2C1%29-functor'>n-homogeneous (∞,1)-functor</a></p>
1322 </li>
1323 </ul>
1324 </li>
1325
1326 <li>
1327 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a></p>
1328
1329 <ul>
1330 <li>
1331 <p><a class='existingWikiWord' href='/nlab/show/analytic+%28%E2%88%9E%2C1%29-functor'>analytic (∞,1)-functor</a></p>
1332 </li>
1333
1334 <li>
1335 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie+spectral+sequence'>Goodwillie spectral sequence</a></p>
1336 </li>
1337 </ul>
1338 </li>
1339 </ul>
1340 </div>
1341
1342 <h4 id='mapping_space'>Mapping space</h4>
1343
1344 <div class='hide'>
1345 <p><strong><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping space</a></strong></p>
1346
1347 <h3 id='general_abstract'>General abstract</h3>
1348
1349 <ul>
1350 <li>
1351 <p><a class='existingWikiWord' href='/nlab/show/hom-set'>hom-set</a>, <a class='existingWikiWord' href='/nlab/show/hom-object'>hom-object</a>, <a class='existingWikiWord' href='/nlab/show/internal+hom'>internal hom</a>, <a class='existingWikiWord' href='/nlab/show/exponential+object'>exponential object</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categorical+hom-space'>derived hom-space</a></p>
1352 </li>
1353
1354 <li>
1355 <p><a class='existingWikiWord' href='/nlab/show/loop+space+object'>loop space object</a>, <a class='existingWikiWord' href='/nlab/show/free+loop+space+object'>free loop space object</a>, <a class='existingWikiWord' href='/nlab/show/derived+loop+space'>derived loop space</a></p>
1356 </li>
1357 </ul>
1358
1359 <h3 id='topology'>Topology</h3>
1360
1361 <ul>
1362 <li>
1363 <p><a class='existingWikiWord' href='/nlab/show/topology+of+mapping+spaces'>topology of mapping spaces</a></p>
1364
1365 <ul>
1366 <li><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>compact-open topology</a></li>
1367 </ul>
1368 </li>
1369
1370 <li>
1371 <p><a class='existingWikiWord' href='/nlab/show/evaluation+fibration+of+mapping+spaces'>evaluation fibration of mapping spaces</a></p>
1372 </li>
1373
1374 <li>
1375 <p><a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/free+loop+space'>free loop space</a></p>
1376 </li>
1377 </ul>
1378
1379 <h3 id='differential_topology'>Differential topology</h3>
1380
1381 <ul>
1382 <li>
1383 <p><a class='existingWikiWord' href='/nlab/show/differential+topology+of+mapping+spaces'>differential topology of mapping spaces</a></p>
1384
1385 <ul>
1386 <li><a class='existingWikiWord' href='/nlab/show/C-infinity+topology'>C-k topology</a></li>
1387 </ul>
1388 </li>
1389
1390 <li>
1391 <p><a class='existingWikiWord' href='/nlab/show/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a></p>
1392
1393 <ul>
1394 <li><a class='existingWikiWord' href='/nlab/show/tangent+spaces+of+mapping+spaces'>tangent spaces of mapping spaces</a></li>
1395 </ul>
1396 </li>
1397
1398 <li>
1399 <p><a class='existingWikiWord' href='/nlab/show/smooth+loop+space'>smooth loop space</a></p>
1400 </li>
1401 </ul>
1402
1403 <h3 id='stable_homotopy_theory_2'>Stable homotopy theory</h3>
1404
1405 <ul>
1406 <li><a class='existingWikiWord' href='/nlab/show/function+spectrum'>mapping spectrum</a></li>
1407 </ul>
1408 <div>
1409 <p>
1410 <a href='/nlab/edit/mapping+space+-+contents'>Edit this sidebar</a>
1411 </p>
1412 </div></div>
1413 </div>
1414 </div>
1415
1416 <h1 id='contents_2'>Contents</h1>
1417 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#Definition'>Definition</a></li><li><a href='#Statements'>Statements</a><ul><li><a href='#prelude_equivalence_to_the_infinite_configuration_space'>Prelude: Equivalence to the infinite configuration space</a></li><li><a href='#StableSplittings'>Stable splitting of mapping spaces</a></li><li><a href='#InTermsOfGoodwillieTowers'>In terms of Goodwillie-Taylor towers</a></li><li><a href='#lax_closed_structure_on_'>Lax closed structure on <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>\Sigma^\infty</annotation></semantics></math></a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div>
1418 <h2 id='idea'>Idea</h2>
1419
1420 <p>The <a class='existingWikiWord' href='/nlab/show/stabilization'>stabilization</a>/<a class='existingWikiWord' href='/nlab/show/suspension+spectrum'>suspension spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma^\infty Maps(X,A)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping spaces</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(X,A)</annotation></semantics></math> between suitable <a class='existingWikiWord' href='/nlab/show/CW+complex'>CW-complexes</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>X, A</annotation></semantics></math> happens to decompose as a <a class='existingWikiWord' href='/nlab/show/direct+sum'>direct sum</a> of <a class='existingWikiWord' href='/nlab/show/spectrum'>spectra</a> (a <a class='existingWikiWord' href='/nlab/show/wedge+sum'>wedge sum</a>) in a useful way, related to the expression of the <a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie derivatives</a> of the functor <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(X,-)</annotation></semantics></math> and often expressible in terms of the <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration spaces</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p>
1421
1422 <h2 id='Definition'>Definition</h2>
1423
1424 <p>The stable splitting of mapping spaces discussed <a href='#StableSplittings'>below</a> have summands given by <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration spaces of points</a>, or generalizations thereof. To be self-contained, we recall the relevant definitions here.</p>
1425
1426 <p>The following Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a> is not the most general definition of <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration spaces of points</a> that one may consider in this context, instead it is streamlined to certain applications. See Remark <a class='maruku-ref' href='#ComparisonToNotationInLiterature'>1</a> below for comparison of notation used here to notation used elsewhere.</p>
1427
1428 <div class='num_defn' id='ConfigurationSpacesOfnPoints'>
1429 <h6 id='definition_2'>Definition</h6>
1430
1431 <p><strong>(<a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration spaces of points</a>)</strong></p>
1432
1433 <p>Let <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a>, possibly with <a class='existingWikiWord' href='/nlab/show/manifold+with+boundary'>boundary</a>.</p>
1434
1435 <p>For <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, the <em><strong>configuration space of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> distinguishable points</strong> in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> disappearing at the boundary</em> is the <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a></p>
1436 <div class='maruku-equation' id='eq:DistinguishableConfigurationSpaceJustForX'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi mathvariant='normal'>Conf</mi> <mi>n</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1437
1438 \mathrm{Conf}^{ord}_{n}(X)
1439 \;\coloneqq\;
1440 \big(
1441 X^n \setminus \mathbf{\Delta}_X^n
1442 \big)
1443 / \partial(X^n)
1444
1445 </annotation></semantics></math></div>
1446 <p>which is the <a class='existingWikiWord' href='/nlab/show/complement'>complement</a> of the <a class='existingWikiWord' href='/nlab/show/fat+diagonal'>fat diagonal</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo>≔</mo><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><msup><mi>x</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo><mo>∈</mo><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy='false'>|</mo><munder><mo>∃</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo stretchy='false'>(</mo><msup><mi>x</mi> <mi>i</mi></msup><mo>=</mo><msup><mi>x</mi> <mi>j</mi></msup><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\mathbf{\Delta}_X^n \coloneqq \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}</annotation></semantics></math> inside the <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/product+topological+space'>product space</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with itself, followed by <a class='existingWikiWord' href='/nlab/show/quotient+space'>collapsing</a> any configurations with elements on the <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to a common <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>base point</a>.</p>
1447
1448 <p>Then the <em><strong>configuration space of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> in-distinguishable points</strong> in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the further <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient topological space</a></em></p>
1449 <div class='maruku-equation' id='eq:ConfigurationSpaceJustForX'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><msubsup><mi>Conf</mi> <mi>n</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msub><mi>Σ</mi> <mi>n</mi></msub><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mo stretchy='false'>/</mo><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'>
1450
1451 \mathrm{Conf}_{n}(X)
1452 \;\coloneqq\;
1453 Conf_n^{ord}(X)/\Sigma_n
1454 \;=\;
1455 \Big(
1456 \big(
1457 X^n \setminus \mathbf{\Delta}_X^n
1458 \big)
1459 / \partial(X^n)
1460 \Big)
1461 /\Sigma(n)
1462 \,,
1463
1464 </annotation></semantics></math></div>
1465 <p>where <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma(n)</annotation></semantics></math> denotes the evident <a class='existingWikiWord' href='/nlab/show/action'>action</a> of the <a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetric group</a> by <a class='existingWikiWord' href='/nlab/show/permutation'>permutation</a> of factors of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> inside <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>X^n</annotation></semantics></math>.</p>
1466
1467 <p>More generally, let <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> be another <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a>, possibly with <a class='existingWikiWord' href='/nlab/show/manifold+with+boundary'>boundary</a>. For <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, the <em><strong>configuration space of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> points</strong> in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math> vanishing at the boundary and distinct as points in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></em> is the <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a></p>
1468 <div class='maruku-equation' id='eq:ConfigurationSpaceWithXAndY'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>Y</mi> <mi>n</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>Y</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mo stretchy='false'>/</mo><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1469
1470 \mathrm{Conf}_{n}(X,Y)
1471 \;\coloneqq\;
1472 \Big(
1473 \big(
1474 (
1475 X^n \setminus \mathbf{\Delta}_X^n
1476 )
1477 \times
1478 Y^n
1479 \big)
1480 / \partial(X^n \times Y^n)
1481 \Big)
1482 /\Sigma(n)
1483
1484 </annotation></semantics></math></div>
1485 <p>where now <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma(n)</annotation></semantics></math> denotes the evident <a class='existingWikiWord' href='/nlab/show/action'>action</a> of the <a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetric group</a> by <a class='existingWikiWord' href='/nlab/show/permutation'>permutation</a> of factors of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math> inside <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>Y</mi> <mi>n</mi></msup><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msup><mo stretchy='false'>)</mo> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>X^n \times Y^n \simeq (X \times Y)^n</annotation></semantics></math>.</p>
1486
1487 <p>This more general definition reduces to the previous case for <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>=</mo><mo>*</mo><mo>≔</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding='application/x-tex'>Y = \ast \coloneqq \mathbb{R}^0</annotation></semantics></math> being the point:</p>
1488 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1489 \mathrm{Conf}_n(X)
1490 \;=\;
1491 \mathrm{Conf}_n(X,\ast)
1492 \,.
1493
1494 </annotation></semantics></math></div>
1495 <p>Finally the <em><strong>configuration space of an arbitrary number of points</strong> in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math> vanishing at the boundary and distinct already as points of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></em> is the <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient topological space</a> of the <a class='existingWikiWord' href='/nlab/show/disjoint+union+topological+space'>disjoint union space</a></p>
1496 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Conf</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>n</mi><mo>∈</mo><mi>𝕟</mi></mrow></munder><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>Y</mi> <mi>k</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>
1497 Conf\left( X, Y\right)
1498 \;\coloneqq\;
1499 \left(
1500 \underset{n \in \mathbb{n}}{\sqcup}
1501 \big(
1502 (
1503 X^n \setminus \mathbf{\Delta}_X^n
1504 )
1505 \times
1506 Y^k
1507 \big)
1508 /\Sigma(n)
1509 \right)/\sim
1510
1511 </annotation></semantics></math></div>
1512 <p>by the <a class='existingWikiWord' href='/nlab/show/equivalence+relation'>equivalence relation</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> given by</p>
1513 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace'></mspace><mo>∼</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo>⇔</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>∈</mo><mo>∂</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1514 \big(
1515 (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n)
1516 \big)
1517 \;\sim\;
1518 \big(
1519 (x_1, y_1), \cdots, (x_{n-1}, y_{n-1})
1520 \big)
1521 \;\;\;\; \Leftrightarrow
1522 \;\;\;\; (x_n, y_n) \in \partial (X \times Y)
1523 \,.
1524
1525 </annotation></semantics></math></div>
1526 <p>This is naturally a <a class='existingWikiWord' href='/nlab/show/filtered+topological+space'>filtered topological space</a> with filter stages</p>
1527 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></munder><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>k</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>k</mi></msubsup><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>Y</mi> <mi>k</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mi>Σ</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>/</mo><mo>∼</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1528 Conf_{\leq n}\left( X, Y\right)
1529 \;\coloneqq\;
1530 \left(
1531 \underset{k \in \{1, \cdots, n\}}{\sqcup}
1532 \big(
1533 (
1534 X^k \setminus \mathbf{\Delta}_X^k
1535 )
1536 \times
1537 Y^k
1538 \big)
1539 /\Sigma(k)
1540 \right)/\sim
1541 \,.
1542
1543 </annotation></semantics></math></div>
1544 <p>The corresponding <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient topological spaces</a> of the filter stages reproduces the above configuration spaces of a fixed number of points:</p>
1545 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msub><mi>Conf</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msub><mi>Conf</mi> <mrow><mo>≤</mo><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1546 Conf_n(X,Y)
1547 \;\simeq\;
1548 Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y)
1549 \,.
1550
1551 </annotation></semantics></math></div></div>
1552
1553 <div class='num_remark' id='ComparisonToNotationInLiterature'>
1554 <h6 id='remark'>Remark</h6>
1555
1556 <p><strong>(comparison to notation in the literature)</strong></p>
1557
1558 <p>The above Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a> is less general but possibly more suggestive than what is considered for instance in <a href='#Boedigheimer87'>Bödigheimer 87</a>. Concretely, we have the following translations of notation:</p>
1559 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mtext> here: </mtext></mtd> <mtd></mtd> <mtd><mrow><mtable><mtr><mtd><mtext> Segal 73,</mtext></mtd></mtr> <mtr><mtd><mtext> Snaith 74</mtext><mo>:</mo></mtd></mtr></mtable></mrow></mtd> <mtd></mtd> <mtd><mtext> Bödigheimer 87: </mtext></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>Conf</mi><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>C</mi> <mi>d</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>C</mi><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>∅</mi><mo>;</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>)</mo></mrow></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>F</mi> <mi>n</mi></msub><msub><mi>C</mi> <mi>d</mi></msub><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msub><mi>F</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>C</mi> <mi>d</mi></msub><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>D</mi> <mi>n</mi></msub><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>∅</mi><mo>;</mo><msup><mi>S</mi> <mn>0</mn></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>F</mi> <mi>n</mi></msub><msub><mi>C</mi> <mi>d</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msub><mi>F</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>C</mi> <mi>d</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>D</mi> <mi>n</mi></msub><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>∅</mi><mo>;</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>D</mi> <mi>n</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>∂</mo><mi>X</mi><mo>;</mo><msup><mi>S</mi> <mn>0</mn></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><msub><mi mathvariant='normal'>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>D</mi> <mi>n</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>∂</mo><mi>X</mi><mo>;</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
1560 \array{
1561 \text{ here: }
1562 &amp;&amp;
1563 \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: }
1564 &amp;&amp;
1565 \text{ Bödigheimer 87: }
1566 \\
1567 \\
1568 Conf(\mathbb{R}^d,Y)
1569 &amp;=&amp;
1570 C_d( Y/\partial Y )
1571 &amp;=&amp;
1572 C( \mathbb{R}^d, \emptyset; Y )
1573 \\
1574 \mathrm{Conf}_n\left( \mathbb{R}^d \right)
1575 &amp; = &amp;
1576 F_n C_d( S^0 ) / F_{n-1} C_d( S^0 )
1577 &amp; = &amp;
1578 D_n\left( \mathbb{R}^d, \emptyset; S^0 \right)
1579 \\
1580 \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right)
1581 &amp; = &amp;
1582 F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y )
1583 &amp; = &amp;
1584 D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right)
1585 \\
1586 \mathrm{Conf}_n( X ) &amp;&amp; &amp;=&amp; D_n\left( X, \partial X; S^0 \right)
1587 \\
1588 \mathrm{Conf}_n( X, Y ) &amp;&amp; &amp;=&amp; D_n\left( X, \partial X; Y/\partial Y \right)
1589 }
1590
1591 </annotation></semantics></math></div>
1592 <p>Notice here that when <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> happens to have <a class='existingWikiWord' href='/nlab/show/empty+space'>empty</a> <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a>, <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∂</mo><mi>Y</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\partial Y = \emptyset</annotation></semantics></math>, then the <a class='existingWikiWord' href='/nlab/show/pushout'>pushout</a></p>
1593 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo>≔</mo><mi>Y</mi><munder><mo>⊔</mo><mrow><mo>∂</mo><mi>Y</mi></mrow></munder><mo>*</mo></mrow><annotation encoding='application/x-tex'>
1594 Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast
1595
1596 </annotation></semantics></math></div>
1597 <p>is <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> with a <a href='pointed+topological+space#ForgettingAndAdjoiningBasepoints'>disjoint basepoint attached</a>. Notably for <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>Y =\ast</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/point+space'>point space</a>, we have that</p>
1598 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo stretchy='false'>/</mo><mo>∂</mo><mo>*</mo><mo>=</mo><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding='application/x-tex'>
1599 \ast/\partial \ast = S^0
1600
1601 </annotation></semantics></math></div>
1602 <p>is the <a class='existingWikiWord' href='/nlab/show/0-sphere'>0-sphere</a>.</p>
1603 </div>
1604
1605 <h2 id='Statements'>Statements</h2>
1606
1607 <h3 id='prelude_equivalence_to_the_infinite_configuration_space'>Prelude: Equivalence to the infinite configuration space</h3>
1608
1609 <p>First recall the following equivalence already before <a class='existingWikiWord' href='/nlab/show/stabilization'>stabilization</a>:</p>
1610
1611 <div class='num_prop' id='ScanningMapEquivalenceOverCartesianSpace'>
1612 <h6 id='proposition'>Proposition</h6>
1613
1614 <p>For</p>
1615
1616 <ol>
1617 <li>
1618 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>d \in \mathbb{N}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>d \geq 1</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/natural+number'>natural number</a> with <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> denoting the <a class='existingWikiWord' href='/nlab/show/cartesian+space'>Cartesian space</a>/<a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclidean space</a> of that <a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a>,</p>
1619 </li>
1620
1621 <li>
1622 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a>, with <a class='existingWikiWord' href='/nlab/show/inhabited+set'>non-empty</a> <a class='existingWikiWord' href='/nlab/show/manifold+with+boundary'>boundary</a> so that <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y / \partial Y</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/connected+space'>connected</a>,</p>
1623 </li>
1624 </ol>
1625
1626 <p>the <a class='existingWikiWord' href='/nlab/show/cohomotopy+charge+map'>scanning map</a> constitutes a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a></p>
1627 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Conf</mi><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo>)</mo></mrow><mover><mo>⟶</mo><mi>scan</mi></mover><msup><mi>Ω</mi> <mi>d</mi></msup><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1628 Conf\left(
1629 \mathbb{R}^d, Y
1630 \right)
1631 \overset{scan}{\longrightarrow}
1632 \Omega^d \Sigma^d (Y/\partial Y)
1633
1634 </annotation></semantics></math></div>
1635 <p>between</p>
1636
1637 <ol>
1638 <li>
1639 <p>the configuration space of arbitrary points in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d \times Y</annotation></semantics></math> vanishing at the boundary (Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a>)</p>
1640 </li>
1641
1642 <li>
1643 <p>the <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</a> of the <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/reduced+suspension'>reduced suspension</a> of the <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient space</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y / \partial Y</annotation></semantics></math> (regarded as a <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed topological space</a> with basepoint <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\partial Y]</annotation></semantics></math>).</p>
1644 </li>
1645 </ol>
1646
1647 <p>In particular when <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi>𝔻</mi> <mi>k</mi></msup></mrow><annotation encoding='application/x-tex'>Y = \mathbb{D}^k</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/ball'>closed ball</a> of <a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k \geq 1</annotation></semantics></math> this gives a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a></p>
1648 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Conf</mi><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>𝔻</mi> <mi>k</mi></msup><mo>)</mo></mrow><mover><mo>⟶</mo><mi>scan</mi></mover><msup><mi>Ω</mi> <mi>d</mi></msup><msup><mi>S</mi> <mrow><mi>d</mi><mo>+</mo><mi>k</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>
1649 Conf\left(
1650 \mathbb{R}^d, \mathbb{D}^k
1651 \right)
1652 \overset{scan}{\longrightarrow}
1653 \Omega^d S^{ d + k }
1654
1655 </annotation></semantics></math></div>
1656 <p>with the <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</a> of the <a class='existingWikiWord' href='/nlab/show/sphere'>(d+k)-sphere</a>.</p>
1657 </div>
1658
1659 <p>(<a href='#May72'>May 72, Theorem 2.7</a>, <a href='#Segal73'>Segal 73, Theorem 3</a>)</p>
1660
1661 <h3 id='StableSplittings'>Stable splitting of mapping spaces</h3>
1662
1663 <div class='num_prop' id='StableSplittingOfMappingSpacesOutOfEuclideanSpace'>
1664 <h6 id='proposition_2'>Proposition</h6>
1665
1666 <p><strong>(<a class='existingWikiWord' href='/nlab/show/stable+splitting+of+mapping+spaces'>stable splitting of mapping spaces</a> out of <a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclidean space</a>/<a class='existingWikiWord' href='/nlab/show/sphere'>n-spheres</a>)</strong></p>
1667
1668 <p>For</p>
1669
1670 <ol>
1671 <li>
1672 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>d \in \mathbb{N}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>d \geq 1</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/natural+number'>natural number</a> with <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> denoting the <a class='existingWikiWord' href='/nlab/show/cartesian+space'>Cartesian space</a>/<a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclidean space</a> of that <a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a>,</p>
1673 </li>
1674
1675 <li>
1676 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a>, with <a class='existingWikiWord' href='/nlab/show/inhabited+set'>non-empty</a> <a class='existingWikiWord' href='/nlab/show/manifold+with+boundary'>boundary</a> so that <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y / \partial Y</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/connected+space'>connected</a>,</p>
1677 </li>
1678 </ol>
1679
1680 <p>there is a <a class='existingWikiWord' href='/nlab/show/stable+weak+homotopy+equivalence'>stable weak homotopy equivalence</a></p>
1681 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Conf</mi><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><munder><mo>⊕</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1682 \Sigma^\infty Conf(\mathbb{R}^d, Y)
1683 \overset{\simeq}{\longrightarrow}
1684 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
1685
1686 </annotation></semantics></math></div>
1687 <p>between</p>
1688
1689 <ol>
1690 <li>
1691 <p>the <a class='existingWikiWord' href='/nlab/show/suspension+spectrum'>suspension spectrum</a> of the <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration space</a> of an arbitrary number of points in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d \times Y</annotation></semantics></math> vanishing at the boundary and distinct already as points of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> (Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a>)</p>
1692 </li>
1693
1694 <li>
1695 <p>the <a class='existingWikiWord' href='/nlab/show/direct+sum'>direct sum</a> (hence: <a class='existingWikiWord' href='/nlab/show/wedge+sum'>wedge sum</a>) of <a class='existingWikiWord' href='/nlab/show/suspension+spectrum'>suspension spectra</a> of the <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration spaces</a> of a fixed number of points in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d \times Y</annotation></semantics></math>, vanishing at the boundary and distinct already as points in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> (also Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a>).</p>
1696 </li>
1697 </ol>
1698
1699 <p>Combined with the <a class='existingWikiWord' href='/nlab/show/stabilization'>stabilization</a> of the <a class='existingWikiWord' href='/nlab/show/cohomotopy+charge+map'>scanning map</a> <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a> from Prop. <a class='maruku-ref' href='#ScanningMapEquivalenceOverCartesianSpace'>1</a> this yields a <a class='existingWikiWord' href='/nlab/show/stable+weak+homotopy+equivalence'>stable weak homotopy equivalence</a></p>
1700 <div class='maruku-equation' id='eq:StableSplittingOfMappingSpacesOutOfSphere'><span class='maruku-eq-number'>(4)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Maps</mi> <mi>cp</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Ω</mi> <mi>d</mi></msup><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><munderover><mo>⟶</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>scan</mi></mrow><mo>≃</mo></munderover><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Conf</mi><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><munder><mo>⊕</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1701
1702 Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
1703 =
1704 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
1705 =
1706 \Omega^d \Sigma^d (Y/\partial Y)
1707 \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow}
1708 \Sigma^\infty Conf(\mathbb{R}^d, Y)
1709 \overset{\simeq}{\longrightarrow}
1710 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
1711
1712 </annotation></semantics></math></div>
1713 <p>between the latter direct sum and the <a class='existingWikiWord' href='/nlab/show/suspension+spectrum'>suspension spectrum</a> of the <a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping space</a> of pointed <a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous functions</a> from the <a class='existingWikiWord' href='/nlab/show/sphere'>d-sphere</a> to the <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/reduced+suspension'>reduced suspension</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y / \partial Y</annotation></semantics></math>.</p>
1714 </div>
1715
1716 <p>(<a href='#Snaith74'>Snaith 74, theorem 1.1</a>, <a href='#Boedigheimer87'>Bödigheimer 87, Example 2</a>)</p>
1717
1718 <p>In fact by <a href='#Boedigheimer87'>Bödigheimer 87, Example 5</a> this equivalence still holds with <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> treated on the same footing as <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math>, hence with <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Conf_n(\mathbb{R}^d, Y)</annotation></semantics></math> on the right replaced by <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Conf_n(\mathbb{R}^d \times Y)</annotation></semantics></math> in the well-adjusted notation of Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a>:</p>
1719 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Maps</mi> <mi>cp</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><munder><mo>⊕</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1720 Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
1721 =
1722 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
1723 \overset{\simeq}{\longrightarrow}
1724 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
1725
1726 </annotation></semantics></math></div>
1727 <h3 id='InTermsOfGoodwillieTowers'>In terms of Goodwillie-Taylor towers</h3>
1728
1729 <p>We discuss the interpretation of the above stable splitting of mapping spaces from the point of view of <a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie calculus</a>, following <a href='#Arone99'>Arone 99, p. 1-2</a>, <a href='#Goodwillie03'>Goodwillie 03, p. 6</a>.</p>
1730
1731 <p>Observe that the <a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration space of points</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Conf_n(X,Y)</annotation></semantics></math> from Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoints'>1</a>, given by the formula <a class='maruku-eqref' href='#eq:ConfigurationSpaceWithXAndY'>(3)</a></p>
1732 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>Y</mi> <mi>n</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>Y</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mo stretchy='false'>/</mo><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1733 Conf_n(X,Y)
1734 \;\coloneqq\;
1735 \Big(
1736 \big(
1737 (
1738 X^n \setminus \mathbf{\Delta}_X^n
1739 )
1740 \times
1741 Y^n
1742 \big)
1743 / \partial(X^n \times Y^n)
1744 \Big)
1745 /\Sigma(n)
1746
1747 </annotation></semantics></math></div>
1748 <p>is the <a class='existingWikiWord' href='/nlab/show/quotient+object'>quotient</a> by the <a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetric group</a>-<a class='existingWikiWord' href='/nlab/show/action'>action</a> of the <em><a class='existingWikiWord' href='/nlab/show/smash+product'>smash product</a></em> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>∧</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><msup><mo stretchy='false'>)</mo> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>Conf_n(X) \wedge (Y/\partial Y)^n</annotation></semantics></math> of the plain Configuration space <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Conf_n(X)</annotation></semantics></math> <a class='maruku-eqref' href='#eq:ConfigurationSpaceJustForX'>(2)</a> (regarded as a <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed topological space</a> with basepoint the class of the <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>[</mo><mo>∂</mo><mrow><mo>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>)</mo></mrow><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'>\left[\partial\left(X^n\right)\right]</annotation></semantics></math>) with the analogous <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed topological space</a> given by <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, the latter in fact being (since here we do not form the <a class='existingWikiWord' href='/nlab/show/complement'>complement</a> by the <a class='existingWikiWord' href='/nlab/show/fat+diagonal'>fat diagonal</a>) an <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-fold <a class='existingWikiWord' href='/nlab/show/smash+product'>smash product</a> itself:</p>
1749 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Y</mi> <mrow><msub><mo>×</mo> <mi>n</mi></msub></mrow></msup><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>Y</mi> <mrow><msub><mo>×</mo> <mi>n</mi></msub></mrow></msup><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><msup><mo stretchy='false'>)</mo> <mrow><msub><mo>∧</mo> <mi>n</mi></msub></mrow></msup><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1750 Y^{\times_n}/\partial (Y^{\times_n})
1751 \;\simeq\;
1752 ( Y/\partial Y )^{\wedge_n}
1753 \,.
1754
1755 </annotation></semantics></math></div>
1756 <p>Hence in summary:</p>
1757 <div class='maruku-equation' id='eq:ConfSplitsAsSmashProduct'><span class='maruku-eq-number'>(5)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msubsup><mi>Conf</mi> <mi>n</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mrow><mo>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo>)</mo></mrow> <mrow><msub><mo>∧</mo> <mi>n</mi></msub></mrow></msup><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'>
1758
1759 Conf_n(X, Y)
1760 \;\simeq\;
1761 Conf^{ord}_n(X) \wedge_{\Sigma(n)} \left( Y/\partial Y \right)^{\wedge_n}
1762 \,,
1763
1764 </annotation></semantics></math></div>
1765 <p>where</p>
1766 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Conf</mi> <mi>n</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mrow><mo>(</mo><msup><mi>X</mi> <mrow><msub><mo>×</mo> <mi>n</mi></msub></mrow></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>X</mi> <mi>n</mi></msubsup><mo>)</mo></mrow><mo stretchy='false'>/</mo><mo>∂</mo><mo stretchy='false'>(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1767 Conf_n^{ord}(X)
1768 \;\coloneqq\;
1769 \left(
1770 X^{\times_n} \setminus \mathbf{\Delta}_X^n
1771 \right)/ \partial(X^n)
1772
1773 </annotation></semantics></math></div>
1774 <p>is the ordered configuration space <a class='maruku-eqref' href='#eq:DistinguishableConfigurationSpaceJustForX'>(1)</a>.</p>
1775
1776 <p>This construction, regarded as a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a> from <a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed topological spaces</a> to <a class='existingWikiWord' href='/nlab/show/spectrum'>spectra</a></p>
1777 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Spectra</mi></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mi>n</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mi>Z</mi> <mrow><msub><mo>∧</mo> <mi>n</mi></msub></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
1778 \array{
1779 Top^{\ast/}
1780 &amp;\longrightarrow&amp;
1781 Spectra
1782 \\
1783 Z
1784 &amp;\mapsto&amp;
1785 \Sigma^\infty Conf^{ord}_n(X) \wedge_{\Sigma(n)} Z^{\wedge_n}
1786 }
1787
1788 </annotation></semantics></math></div>
1789 <p>is an <a class='existingWikiWord' href='/nlab/show/n-homogeneous+%28%E2%88%9E%2C1%29-functor'>n-homogeneous (∞,1)-functor</a> in the sense of <a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie calculus</a>, and hence the partial <a class='existingWikiWord' href='/nlab/show/wedge+sum'>wedge sums</a> as <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> ranges</p>
1790 <div class='maruku-equation' id='eq:IdentifyingTheGoodwillieTaylorStage'><span class='maruku-eq-number'>(6)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi><mspace width='thickmathspace'></mspace><mo>↦</mo><mspace width='thickmathspace'></mspace><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⨁</mo><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mo>⋅</mo><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mi>k</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mi>Z</mi> <mrow><msub><mo>∧</mo> <mi>k</mi></msub></mrow></msup></mrow><annotation encoding='application/x-tex'>
1791
1792 Z
1793 \;\mapsto\;
1794 \underset{k \in \{1, \cdot, n\}}{\bigoplus}
1795 \Sigma^\infty Conf^{ord}_k(X) \wedge_{\Sigma(k)} Z^{\wedge_k}
1796
1797 </annotation></semantics></math></div>
1798 <p>are <a class='existingWikiWord' href='/nlab/show/n-excisive+%28%E2%88%9E%2C1%29-functor'>n-excisive (∞,1)-functors</a>. Moreover, by the stable splitting of mapping spaces <a class='maruku-eqref' href='#eq:StableSplittingOfMappingSpacesOutOfSphere'>(4)</a> of Prop. <a class='maruku-ref' href='#StableSplittingOfMappingSpacesOutOfEuclideanSpace'>2</a>, there is a <a class='existingWikiWord' href='/nlab/show/projection'>projection</a> morphism onto the first <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/wedge+sum'>wedge summands</a></p>
1799 <div class='maruku-equation' id='eq:ProjectionMaps'><span class='maruku-eq-number'>(7)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Maps</mi> <mi>cp</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mi>Z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mi>Z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><munder><mo>⊕</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mi>k</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mi>Z</mi> <mrow><msub><mo>∧</mo> <mi>k</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo maxsize='1.8em' minsize='1.8em'>↓</mo><msup><mrow></mrow> <mpadded width='0'><mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⨁</mo><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mo>⋅</mo><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mi>k</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mi>Z</mi> <mrow><msub><mo>∧</mo> <mi>k</mi></msub></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
1800
1801 \array{
1802 Maps_{cp}(\mathbb{R}^d, \Sigma^d Z)
1803 &amp;=&amp;
1804 Maps^{\ast/}( S^d, \Sigma^d Z)
1805 &amp;\simeq&amp;
1806 \underset{k \in \mathbb{N}}{\oplus}
1807 \Sigma^\infty Conf^{ord}_k(\mathbb{R}^d) \wedge_{\Sigma(k)} Z^{\wedge_k}
1808 \\
1809 &amp;&amp;
1810 &amp;&amp;
1811 \Big\downarrow {}^{\mathrlap{ p_n }}
1812 \\
1813 &amp;&amp;
1814 &amp;&amp;
1815 \underset{k \in \{1, \cdot, n\}}{\bigoplus}
1816 \Sigma^\infty Conf^{ord}_k( \mathbb{R}^d ) \wedge_{\Sigma(k)} Z^{\wedge_k}
1817 }
1818
1819 </annotation></semantics></math></div>
1820 <p>and this is <a class='existingWikiWord' href='/nlab/show/n-connected+object+of+an+%28infinity%2C1%29-topos'>(n+1)k-connected</a> when <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/n-connected+object+of+an+%28infinity%2C1%29-topos'>k-connected</a>.</p>
1821
1822 <p>By <a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>Goodwillie calculus</a> this means that <a class='maruku-eqref' href='#eq:IdentifyingTheGoodwillieTaylorStage'>(6)</a> are, up to <a class='existingWikiWord' href='/nlab/show/equivalence+in+an+%28infinity%2C1%29-category'>equivalence</a>, the stages</p>
1823 <div class='maruku-equation' id='eq:TheGoodwillieStagesOfTheMappingSpaceFunctor'><span class='maruku-eq-number'>(8)</span><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>P</mi> <mi>n</mi></msub><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>Z</mi><mo>↦</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⨁</mo><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mo>⋅</mo><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></munder><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mi>k</mi> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
1824
1825 P_n Maps^{\ast/}( S^d, \Sigma^d (-))
1826 \;\colon\;
1827 Z \mapsto
1828 \underset{k \in \{1, \cdot, n\}}{\bigoplus}
1829 \Sigma^\infty Conf^{ord}_k(S^d, Z)
1830
1831 </annotation></semantics></math></div>
1832 <p>at <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Z \in Top^{\ast/}</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a> for the <a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping space</a>-functor</p>
1833 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Maps</mi> <mi>cp</mi></msub><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>⟶</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1834 Maps_{cp}(\mathbb{R}^d, \Sigma^d (-))
1835 =
1836 Maps^{\ast/}( S^d, \Sigma^d (-))
1837 \;\colon\;
1838 Top^{\ast/} \longrightarrow Top^{\ast/}
1839 \,.
1840
1841 </annotation></semantics></math></div>
1842 <p>Therefore the stable splitting theorem <a class='maruku-ref' href='#StableSplittingOfMappingSpacesOutOfEuclideanSpace'>2</a> may equivalently be read as expressing the mapping space functor equivalently as the <a class='existingWikiWord' href='/nlab/show/limit'>limit</a> over its <a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a>.</p>
1843
1844 <p>(<a href='#Arone99'>Arone 99, p. 1-2</a>, <a href='#Goodwillie03'>Goodwillie 03, p. 6</a>)</p>
1845
1846 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace'></mspace></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p>
1847
1848 <h3 id='lax_closed_structure_on_'>Lax closed structure on <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>\Sigma^\infty</annotation></semantics></math></h3>
1849
1850 <p>Notice that the first stage in the <a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(S^d, \Sigma^d(-))</annotation></semantics></math> is</p>
1851 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>P</mi> <mn>1</mn></msub><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msubsup><mi>Conf</mi> <mn>1</mn> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><munder><munder><mrow><msubsup><mi>Conf</mi> <mn>1</mn> <mi>ord</mi></msubsup><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy='false'>)</mo></mrow><mo>⏟</mo></munder><mrow><mo>≃</mo><msup><mi>S</mi> <mn>0</mn></msup></mrow></munder><mo>∧</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>Ω</mi> <mi>d</mi></msup><msup><mi>Σ</mi> <mi>d</mi></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Maps</mi><mrow><mo>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
1852 \begin{aligned}
1853 P_1 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y) )
1854 &amp; =
1855 \Sigma^\infty Conf^{ord}_1( \mathbb{R}^d , Y )
1856 \\
1857 &amp; \simeq
1858 \Sigma^\infty
1859 \underset{\simeq S^0}{\underbrace{Conf^{ord}_1( \mathbb{R}^d )}}
1860 \wedge (Y/\partial Y)
1861 \\
1862 &amp; \simeq
1863 \Sigma^\infty (Y/\partial Y)
1864 \\
1865 &amp; \simeq
1866 \Omega^d \Sigma^d \Sigma^\infty (Y/\partial Y)
1867 \\
1868 &amp; \simeq
1869 Maps\left( \Sigma^\infty S^d, \Sigma^d (Y/\partial Y) \right)
1870 \end{aligned}
1871
1872 </annotation></semantics></math></div>
1873 <p>Here in the first step we used <a class='maruku-eqref' href='#eq:TheGoodwillieStagesOfTheMappingSpaceFunctor'>(8)</a>, in the second step we used <a class='maruku-eqref' href='#eq:ConfSplitsAsSmashProduct'>(5)</a>. Under the brace we observe that space of configurations of a single point in <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> is trivially <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> itself, which is <a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible</a> <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>≃</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d \simeq \ast</annotation></semantics></math> and, due to <a class='existingWikiWord' href='/nlab/show/empty+set'>empty</a> <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a> of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math>, contributes a <a class='existingWikiWord' href='/nlab/show/0-sphere'>0-sphere</a>-factor to the <a class='existingWikiWord' href='/nlab/show/smash+product'>smash product</a>, which disappears. In the last last two steps we trivially rewrote the result to exhibit it as a <a class='existingWikiWord' href='/nlab/show/function+spectrum'>mapping spectrum</a>.</p>
1874
1875 <p>Therefore the projection <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>p_1</annotation></semantics></math> <a class='maruku-eqref' href='#eq:ProjectionMaps'>(7)</a> to the first stage of the <a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a> is of the form</p>
1876 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Maps</mi><mrow><mo>(</mo><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo>⟶</mo><mi>Maps</mi><mrow><mo>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mi>d</mi></msup><mo>,</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mi>d</mi></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>/</mo><mo>∂</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
1877 p_1
1878 \;\colon\;
1879 \Sigma^\infty Maps\left( S^d , \Sigma^d (Y /\partial Y) \right)
1880 \longrightarrow
1881 Maps
1882 \left(
1883 \Sigma^\infty S^d, \Sigma^\infty \Sigma^d (Y / \partial Y)
1884 \right)
1885 \,.
1886
1887 </annotation></semantics></math></div>
1888 <p>Since <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>\Sigma^\infty</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/monoidal+functor'>strong monoidal functor</a> (<a href='suspension+spectrum#StrongMonoidalness'>here</a>), there is a canonical comparison morphism of this form, exhibiting the induce <a class='existingWikiWord' href='/nlab/show/closed+functor'>lax closed</a>-structure on <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>\Sigma^\infty</annotation></semantics></math>. Probably <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>p_1</annotation></semantics></math> coincides with that canonical morphism, up to equivalence.</p>
1889
1890 <blockquote>
1891 <p>Does it?</p>
1892 </blockquote>
1893
1894 <h2 id='related_concepts'>Related concepts</h2>
1895
1896 <ul>
1897 <li><a class='existingWikiWord' href='/nlab/show/function+spectrum'>mapping spectrum</a></li>
1898 </ul>
1899
1900 <h2 id='references'>References</h2>
1901
1902 <p>The theorem is originally due to</p>
1903
1904 <ul>
1905 <li id='Snaith74'><a class='existingWikiWord' href='/nlab/show/Victor+Snaith'>Victor Snaith</a>, <em>A stable decomposition of <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e12cd452fffa20b013a4319559b55_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><msup><mi>S</mi> <mi>n</mi></msup><mi>X</mi></mrow><annotation encoding='application/x-tex'>\Omega^n S^n X</annotation></semantics></math></em>, Journal of the London Mathematical Society 7 (1974), 577 - 583 (<a href='https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf'>pdf</a>)</li>
1906 </ul>
1907
1908 <p>using the homotopy equivalence before stabilization due to</p>
1909
1910 <ul>
1911 <li id='May72'>
1912 <p><a class='existingWikiWord' href='/nlab/show/Peter+May'>Peter May</a>, <em>The geometry of iterated loop spaces</em>, Springer 1972 (<a href='https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf'>pdf</a>)</p>
1913 </li>
1914
1915 <li id='Segal73'>
1916 <p><a class='existingWikiWord' href='/nlab/show/Graeme+Segal'>Graeme Segal</a>, <em>Configuration-spaces and iterated loop-spaces</em>, Invent. Math. <strong>21</strong> (1973), 213–221. MR 0331377 (<a href='http://dodo.pdmi.ras.ru/~topology/books/segal.pdf'>pdf</a>)</p>
1917 </li>
1918 </ul>
1919
1920 <p>An alternative proof is due to</p>
1921
1922 <ul>
1923 <li id='Cohen80'><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen</a>, <em>Stable proof of stable splittings</em>, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 (<a href='https://doi.org/10.1017/S030500410005742X'>doi:10.1017/S030500410005742X</a>, <a href='https://www.cambridge.org/core/services/aop-cambridge-core/content/view/247D9F951F8AB99000E4FF6CB2DB9EA2/S030500410005742Xa.pdf/div-class-title-stable-proofs-of-stable-splittings-div.pdf'>pdf</a>)</li>
1924 </ul>
1925
1926 <p>Review and generalization is due to</p>
1927
1928 <ul>
1929 <li id='Boedigheimer87'><a class='existingWikiWord' href='/nlab/show/Carl-Friedrich+B%C3%B6digheimer'>Carl-Friedrich Bödigheimer</a>, <em>Stable splittings of mapping spaces</em>, Algebraic topology. Springer 1987. 174-187 (<a href='http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf'>pdf</a>)</li>
1930 </ul>
1931
1932 <p>Interpretation in terms of the <a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'>Goodwillie-Taylor tower</a> of mapping spaces is due to</p>
1933
1934 <ul>
1935 <li id='Arone99'>
1936 <p><a class='existingWikiWord' href='/nlab/show/Gregory+Arone'>Greg Arone</a>, <em>A generalization of Snaith-type filtration</em>, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (<a href='https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf'>pdf</a>)</p>
1937 </li>
1938
1939 <li id='Ching05'>
1940 <p><a class='existingWikiWord' href='/nlab/show/Michael+Ching'>Michael Ching</a>, <em>Calculus of Functors and Configuration Spaces</em>, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (<a href='https://www3.amherst.edu/~mching/Work/skye.pdf'>pdf</a>)</p>
1941 </li>
1942
1943 <li id='Goodwillie03'>
1944 <p><a class='existingWikiWord' href='/nlab/show/Thomas+Goodwillie'>Thomas Goodwillie</a>, p. 6 of <em>Calculus. III. Taylor series</em>, Geom. Topol. 7 (2003), 645–711 (<a href='http://www.msp.warwick.ac.uk/gt/2003/07/p019.xhtml'>journal</a>, <a href='http://arxiv.org/abs/math/0310481'>arXiv:math/0310481</a>))</p>
1945 </li>
1946 </ul>
1947
1948 <p>A proof via <a class='existingWikiWord' href='/nlab/show/nonabelian+Poincar%C3%A9+duality'>nonabelian Poincaré duality</a>:</p>
1949
1950 <ul>
1951 <li>Lauren Bandklayder, <em>Stable splitting of mapping spaces via nonabelian Poincaré duality</em> (<a href='https://arxiv.org/abs/1705.03090'>arxiv:1705.03090</a>)</li>
1952 </ul>
1953
1954 <p>See also:</p>
1955
1956 <ul>
1957 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas Ravenel</a>, <em>What we still don’t understand about loop spaces of spheres</em>, Contemporary Mathematics 1998 (<a href='https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Ravenel_LoopSpacesOfSpheres.pdf' title='pdf'>pdf</a>)</li>
1958 </ul>
1959
1960 <p>
1961 </p> </div>
1962 </content>
1963 </entry>
1964 <entry>
1965 <title type="html">Tarmo Uustalu</title>
1966 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Tarmo+Uustalu"/>
1967 <updated>2021-07-02T00:18:56Z</updated>
1968 <published>2021-07-01T06:30:08Z</published>
1969 <id>tag:ncatlab.org,2021-07-01:nLab,Tarmo+Uustalu</id>
1970 <author>
1971 <name>Dmitri Pavlov</name>
1972 </author>
1973 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Tarmo+Uustalu">
1974 <div xmlns="http://www.w3.org/1999/xhtml">
1975 <p>Tarmo Uustalu is a professor at the Dept. of Computer Science of Reykjavik University. He also has part-time post in the Dept. of Software Science of the Tallinn University of Technology (TUT) as a lead research scientist, taking care of the Lab for High-Assurance Software, in particular the Logic and Semantics Group.</p>
1976
1977 <ul>
1978 <li><a href='https://www.ioc.ee/~tarmo/'>Home page</a></li>
1979 </ul>
1980
1981 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p>
1982
1983 <p>
1984 </p> </div>
1985 </content>
1986 </entry>
1987 <entry>
1988 <title type="html">Monster group</title>
1989 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Monster+group"/>
1990 <updated>2021-07-01T21:40:24Z</updated>
1991 <published>2010-05-19T06:28:51Z</published>
1992 <id>tag:ncatlab.org,2010-05-19:nLab,Monster+group</id>
1993 <author>
1994 <name>Urs Schreiber</name>
1995 </author>
1996 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Monster+group">
1997 <div xmlns="http://www.w3.org/1999/xhtml">
1998 <div class='rightHandSide'>
1999 <div class='toc clickDown' tabindex='0'>
2000 <h3 id='context'>Context</h3>
2001
2002 <h4 id='exceptional_structures'>Exceptional structures</h4>
2003
2004 <div class='hide'>
2005 <p><strong><a class='existingWikiWord' href='/nlab/show/exceptional+structure'>exceptional structures</a></strong>, <a class='existingWikiWord' href='/nlab/show/exceptional+isomorphism'>exceptional isomorphisms</a></p>
2006
2007 <h2 id='examples'>Examples</h2>
2008
2009 <ul>
2010 <li>
2011 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>exceptional finite groups</a></p>
2012
2013 <ul>
2014 <li>
2015 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>monster group</a></p>
2016 </li>
2017
2018 <li>
2019 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu group</a>,</p>
2020 </li>
2021
2022 <li>
2023 <p><a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway group</a></p>
2024 </li>
2025 </ul>
2026 </li>
2027
2028 <li>
2029 <p>exceptional <a class='existingWikiWord' href='/nlab/show/finite+rotation+group'>finite rotation groups</a>:</p>
2030
2031 <ul>
2032 <li>
2033 <p><a class='existingWikiWord' href='/nlab/show/tetrahedral+group'>tetrahedral group</a></p>
2034 </li>
2035
2036 <li>
2037 <p><a class='existingWikiWord' href='/nlab/show/octahedral+group'>octahedral group</a></p>
2038 </li>
2039
2040 <li>
2041 <p><a class='existingWikiWord' href='/nlab/show/icosahedral+group'>icosahedral group</a></p>
2042 </li>
2043 </ul>
2044 </li>
2045
2046 <li>
2047 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+group'>exceptional Lie groups</a></p>
2048
2049 <ul>
2050 <li>
2051 <p><a class='existingWikiWord' href='/nlab/show/G2'>G2</a></p>
2052 </li>
2053
2054 <li>
2055 <p><a class='existingWikiWord' href='/nlab/show/F4'>F4</a></p>
2056 </li>
2057
2058 <li>
2059 <p><a class='existingWikiWord' href='/nlab/show/E6'>E6</a>, <a class='existingWikiWord' href='/nlab/show/E7'>E7</a>, <a class='existingWikiWord' href='/nlab/show/E8'>E8</a></p>
2060 </li>
2061 </ul>
2062
2063 <p>and <a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moody groups</a>:</p>
2064
2065 <ul>
2066 <li><a class='existingWikiWord' href='/nlab/show/E9'>E9</a>, <a class='existingWikiWord' href='/nlab/show/E10'>E10</a>, <a class='existingWikiWord' href='/nlab/show/E11'>E11</a>, …</li>
2067 </ul>
2068 </li>
2069
2070 <li>
2071 <p><a class='existingWikiWord' href='/nlab/show/Dwyer-Wilkerson+H-space'>Dwyer-Wilkerson H-space</a></p>
2072 </li>
2073
2074 <li>
2075 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+algebra'>exceptional Lie algebras</a></p>
2076 </li>
2077
2078 <li>
2079 <p><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>exceptional Jordan algebra</a></p>
2080
2081 <ul>
2082 <li><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>Albert algebra</a></li>
2083 </ul>
2084 </li>
2085
2086 <li>
2087 <p>exceptional <a class='existingWikiWord' href='/nlab/show/Jordan+superalgebra'>Jordan superalgebra</a>, <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mn>10</mn></msub></mrow><annotation encoding='application/x-tex'>K_10</annotation></semantics></math></p>
2088 </li>
2089
2090 <li>
2091 <p><a class='existingWikiWord' href='/nlab/show/Leech+lattice'>Leech lattice</a></p>
2092 </li>
2093
2094 <li>
2095 <p><a class='existingWikiWord' href='/nlab/show/Cayley+plane'>Cayley plane</a></p>
2096 </li>
2097 </ul>
2098
2099 <h2 id='interrelations'>Interrelations</h2>
2100
2101 <ul>
2102 <li>
2103 <p><a class='existingWikiWord' href='/nlab/show/division+algebra+and+supersymmetry'>supersymmetry and division algebras</a></p>
2104 </li>
2105
2106 <li>
2107 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+magic+square'>Freudenthal magic square</a></p>
2108 </li>
2109
2110 <li>
2111 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>moonshine</a></p>
2112
2113 <ul>
2114 <li>
2115 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathieu moonshine</a></p>
2116 </li>
2117
2118 <li>
2119 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral moonshine</a></p>
2120 </li>
2121
2122 <li>
2123 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O&#39;Nan moonshine</a></p>
2124 </li>
2125 </ul>
2126 </li>
2127 </ul>
2128
2129 <h2 id='applications'>Applications</h2>
2130
2131 <ul>
2132 <li>
2133 <p><a class='existingWikiWord' href='/nlab/show/exceptional+geometry'>exceptional geometry</a>, <a class='existingWikiWord' href='/nlab/show/exceptional+generalized+geometry'>exceptional generalized geometry</a>,</p>
2134 </li>
2135
2136 <li>
2137 <p><a class='existingWikiWord' href='/nlab/show/exceptional+field+theory'>exceptional field theory</a></p>
2138 </li>
2139 </ul>
2140
2141 <h2 id='philosophy'>Philosophy</h2>
2142
2143 <ul>
2144 <li><a class='existingWikiWord' href='/nlab/show/universal+exceptionalism'>universal exceptionalism</a></li>
2145 </ul>
2146 </div>
2147
2148 <h4 id='group_theory'>Group Theory</h4>
2149
2150 <div class='hide'>
2151 <p><strong><a class='existingWikiWord' href='/nlab/show/group+theory'>group theory</a></strong></p>
2152
2153 <ul>
2154 <li><a class='existingWikiWord' href='/nlab/show/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-group</a></li>
2155
2156 <li><a class='existingWikiWord' href='/nlab/show/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li>
2157
2158 <li><a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></li>
2159
2160 <li><a class='existingWikiWord' href='/nlab/show/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/infinity-action'>∞-action</a></li>
2161
2162 <li><a class='existingWikiWord' href='/nlab/show/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/infinity-representation'>∞-representation</a></li>
2163
2164 <li><a class='existingWikiWord' href='/nlab/show/progroup'>progroup</a></li>
2165
2166 <li><a class='existingWikiWord' href='/nlab/show/homogeneous+space'>homogeneous space</a></li>
2167 </ul>
2168
2169 <h3 id='classical_groups'>Classical groups</h3>
2170
2171 <ul>
2172 <li>
2173 <p><a class='existingWikiWord' href='/nlab/show/general+linear+group'>general linear group</a></p>
2174 </li>
2175
2176 <li>
2177 <p><a class='existingWikiWord' href='/nlab/show/unitary+group'>unitary group</a></p>
2178
2179 <ul>
2180 <li><a class='existingWikiWord' href='/nlab/show/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/projective+unitary+group'>projective unitary group</a></li>
2181 </ul>
2182 </li>
2183
2184 <li>
2185 <p><a class='existingWikiWord' href='/nlab/show/orthogonal+group'>orthogonal group</a></p>
2186
2187 <ul>
2188 <li><a class='existingWikiWord' href='/nlab/show/special+orthogonal+group'>special orthogonal group</a></li>
2189 </ul>
2190 </li>
2191
2192 <li>
2193 <p><a class='existingWikiWord' href='/nlab/show/symplectic+group'>symplectic group</a></p>
2194 </li>
2195 </ul>
2196
2197 <h3 id='finite_groups'>Finite groups</h3>
2198
2199 <ul>
2200 <li>
2201 <p><a class='existingWikiWord' href='/nlab/show/finite+group'>finite group</a></p>
2202 </li>
2203
2204 <li>
2205 <p><a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/braid+group'>braid group</a></p>
2206 </li>
2207
2208 <li>
2209 <p><a class='existingWikiWord' href='/nlab/show/classification+of+finite+simple+groups'>classification of finite simple groups</a></p>
2210 </li>
2211
2212 <li>
2213 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>sporadic finite simple groups</a></p>
2214
2215 <ul>
2216 <li><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu group</a></li>
2217 </ul>
2218 </li>
2219 </ul>
2220
2221 <h3 id='group_schemes'>Group schemes</h3>
2222
2223 <ul>
2224 <li><a class='existingWikiWord' href='/nlab/show/algebraic+group'>algebraic group</a></li>
2225
2226 <li><a class='existingWikiWord' href='/nlab/show/abelian+variety'>abelian variety</a></li>
2227 </ul>
2228
2229 <h3 id='topological_groups'>Topological groups</h3>
2230
2231 <ul>
2232 <li>
2233 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topological group</a></p>
2234 </li>
2235
2236 <li>
2237 <p><a class='existingWikiWord' href='/nlab/show/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/locally+compact+topological+group'>locally compact topological group</a></p>
2238 </li>
2239
2240 <li>
2241 <p><a class='existingWikiWord' href='/nlab/show/maximal+compact+subgroup'>maximal compact subgroup</a></p>
2242 </li>
2243
2244 <li>
2245 <p><a class='existingWikiWord' href='/nlab/show/string+group'>string group</a></p>
2246 </li>
2247 </ul>
2248
2249 <h3 id='lie_groups'>Lie groups</h3>
2250
2251 <ul>
2252 <li>
2253 <p><a class='existingWikiWord' href='/nlab/show/Lie+group'>Lie group</a></p>
2254 </li>
2255
2256 <li>
2257 <p><a class='existingWikiWord' href='/nlab/show/compact+Lie+group'>compact Lie group</a></p>
2258 </li>
2259
2260 <li>
2261 <p><a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moody group</a></p>
2262 </li>
2263 </ul>
2264
2265 <h3 id='superlie_groups'>Super-Lie groups</h3>
2266
2267 <ul>
2268 <li>
2269 <p><a class='existingWikiWord' href='/nlab/show/supergroup'>super Lie group</a></p>
2270 </li>
2271
2272 <li>
2273 <p><a class='existingWikiWord' href='/nlab/show/super+Euclidean+group'>super Euclidean group</a></p>
2274 </li>
2275 </ul>
2276
2277 <h3 id='higher_groups'>Higher groups</h3>
2278
2279 <ul>
2280 <li>
2281 <p><a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a></p>
2282
2283 <ul>
2284 <li><a class='existingWikiWord' href='/nlab/show/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/strict+2-group'>strict 2-group</a></li>
2285 </ul>
2286 </li>
2287
2288 <li>
2289 <p><a class='existingWikiWord' href='/nlab/show/n-group'>n-group</a></p>
2290 </li>
2291
2292 <li>
2293 <p><a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-group</a></p>
2294
2295 <ul>
2296 <li>
2297 <p><a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a></p>
2298 </li>
2299
2300 <li>
2301 <p><a class='existingWikiWord' href='/nlab/show/crossed+complex'>crossed complex</a></p>
2302 </li>
2303
2304 <li>
2305 <p><a class='existingWikiWord' href='/nlab/show/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p>
2306 </li>
2307
2308 <li>
2309 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></p>
2310 </li>
2311 </ul>
2312 </li>
2313
2314 <li>
2315 <p><a class='existingWikiWord' href='/nlab/show/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/fivebrane+6-group'>fivebrane Lie 6-group</a></p>
2316 </li>
2317 </ul>
2318
2319 <h3 id='cohomology_and_extensions'>Cohomology and Extensions</h3>
2320
2321 <ul>
2322 <li>
2323 <p><a class='existingWikiWord' href='/nlab/show/group+cohomology'>group cohomology</a></p>
2324 </li>
2325
2326 <li>
2327 <p><a class='existingWikiWord' href='/nlab/show/group+extension'>group extension</a>,</p>
2328 </li>
2329
2330 <li>
2331 <p><a class='existingWikiWord' href='/nlab/show/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/Ext'>Ext-group</a></p>
2332 </li>
2333 </ul>
2334
2335 <h3 id='_related_concepts'>Related concepts</h3>
2336
2337 <ul>
2338 <li><a class='existingWikiWord' href='/nlab/show/quantum+group'>quantum group</a></li>
2339 </ul>
2340 <div>
2341 <p>
2342 <a href='/nlab/edit/group+theory+-+contents'>Edit this sidebar</a>
2343 </p>
2344 </div></div>
2345 </div>
2346 </div>
2347
2348 <h1 id='contents'>Contents</h1>
2349 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#history'>History</a></li><li><a href='#presentation'>Presentation</a><ul><li><a href='#via_coxeter_groups'>Via Coxeter groups</a></li><li><a href='#ViaAutomorphisms'>Via automorphisms of a super vertex operator algebra</a></li></ul></li><li><a href='#related_concepts_2'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div>
2350 <h2 id='idea'>Idea</h2>
2351
2352 <p>The <strong>Monster group</strong> <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/finite+group'>finite group</a> that is the largest of the <a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>sporadic finite simple group</a>s. It has <a class='existingWikiWord' href='/nlab/show/order+of+a+group'>order</a></p>
2353 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd></mtd> <mtd><msup><mn>2</mn> <mn>46</mn></msup><mo>⋅</mo><msup><mn>3</mn> <mn>20</mn></msup><mo>⋅</mo><msup><mn>5</mn> <mn>9</mn></msup><mo>⋅</mo><msup><mn>7</mn> <mn>6</mn></msup><mo>⋅</mo><msup><mn>11</mn> <mn>2</mn></msup><mo>⋅</mo><msup><mn>13</mn> <mn>3</mn></msup><mo>⋅</mo><mn>17</mn><mo>⋅</mo><mn>19</mn><mo>⋅</mo><mn>23</mn><mo>⋅</mo><mn>29</mn><mo>⋅</mo><mn>31</mn><mo>⋅</mo><mn>41</mn><mo>⋅</mo><mn>47</mn><mo>⋅</mo><mn>59</mn><mo>⋅</mo><mn>71</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>808017424794512875886459904961710757005754368000000000</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
2354 \begin{aligned}
2355 &amp; 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71
2356 \\
2357 &amp; = 808017424794512875886459904961710757005754368000000000
2358 \end{aligned}
2359
2360 </annotation></semantics></math></div>
2361 <p>and contains all but six (the ‘<a class='existingWikiWord' href='/nlab/show/pariah+group'>pariah groups</a>’) of the other 25 <a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>sporadic finite simple groups</a> as <a class='existingWikiWord' href='/nlab/show/subquotient'>subquotients</a>, called the <em><a class='existingWikiWord' href='/nlab/show/Happy+Family'>Happy Family</a></em>.</p>
2362
2363 <p>See also <a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine</a>.</p>
2364
2365 <h2 id='history'>History</h2>
2366
2367 <p>The Monster group was predicted to exist by <a class='existingWikiWord' href='/nlab/show/Bernd+Fischer'>Bernd Fischer</a> and <a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Griess</a> in 1973, as a <a class='existingWikiWord' href='/nlab/show/simple+group'>simple group</a> containing the <a class='existingWikiWord' href='/nlab/show/Fischer+group'>Fischer groups</a> and some other sporadic simple groups as <a class='existingWikiWord' href='/nlab/show/subquotient'>subquotients</a>. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> and discovered other properties and subgroups, assuming that it existed. In a famous paper</p>
2368
2369 <ul>
2370 <li><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Griess</a>, <em>The Friendly Giant</em> , Inventiones (1982)</li>
2371 </ul>
2372
2373 <p>Griess proved the existence of the largest simple sporadic group. The author constructs “by hand” a non-associative but commutative algebra of dimension 196883, and showed that the <a class='existingWikiWord' href='/nlab/show/automorphism'>automorphism group</a> of this algebra is the conjectured friendly giant/monster simple group. The name “Friendly Giant” for the Monster did not take on.</p>
2374
2375 <p>After Griess found this algebra <a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenkel</a>, <a class='existingWikiWord' href='/nlab/show/James+Lepowsky'>James Lepowsky</a> and Meurman and/or Borcherds showed that the Griess algebra is just the degree 2 part of the infinite dimensional <a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine vertex algebra</a>.</p>
2376
2377 <p>There is a school of thought, going back to at least <a class='existingWikiWord' href='/nlab/show/Israel+Gelfand'>Israel Gelfand</a>, that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see <a href='http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html'>this nCafe post</a>). One version of this, in the case of the Monster (and perhaps for other sporadic groups via <a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine</a> phenomena) is that what we know as the Monster is just a shadow of a <a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a>, as the Monster can be constructed as an automorphism group of a <a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>conformal field theory</a>, a structure rich enough to have a automorphism 2-group(oid) (see <a href='http://golem.ph.utexas.edu/category/2008/10/john_mckay_visits_kent.html#c019440'>this nCafe discussion</a>).</p>
2378
2379 <h2 id='presentation'>Presentation</h2>
2380
2381 <h3 id='via_coxeter_groups'>Via Coxeter groups</h3>
2382
2383 <p>The Monster admits a reasonably succinct description in terms of <a class='existingWikiWord' href='/nlab/show/Coxeter+group'>Coxeter groups</a>. Let <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[n]</annotation></semantics></math> denote the linear <a class='existingWikiWord' href='/nlab/show/graph'>graph</a> with vertices <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>0, 1, \ldots, n</annotation></semantics></math> with an edge between adjacent numbers <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>i, i+1</annotation></semantics></math> and no others. If <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> is the terminal (1-element) graph, there is a map <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>:</mo><mn>1</mn><mo>→</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>0: 1 \to [n]</annotation></semantics></math>, mapping the vertex of <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> to the vertex <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>. Regarding this as an object in the <a class='existingWikiWord' href='/nlab/show/under+category'>undercategory</a> <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>↓</mo><mi>Graph</mi></mrow><annotation encoding='application/x-tex'>1 \downarrow Graph</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mn>443</mn></msub></mrow><annotation encoding='application/x-tex'>Y_{443}</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/coproduct'>coproduct</a> of the three objects <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>:</mo><mn>1</mn><mo>→</mo><mo stretchy='false'>[</mo><mn>4</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>0: 1 \to [4]</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>:</mo><mn>1</mn><mo>→</mo><mo stretchy='false'>[</mo><mn>4</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>0: 1 \to [4]</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>:</mo><mn>1</mn><mo>→</mo><mo stretchy='false'>[</mo><mn>3</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>0: 1 \to [3]</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>↓</mo><mi>Graph</mi></mrow><annotation encoding='application/x-tex'>1 \downarrow Graph</annotation></semantics></math>. This (pointed) graph has 12 elements and is shaped like a <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, with arms of length 4, 4, 3 emanating from a central vertex of valence <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>.</p>
2384
2385 <p>Regard <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mn>443</mn></msub></mrow><annotation encoding='application/x-tex'>Y_{443}</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/Coxeter+group'>Coxeter diagram</a>. The associated <a class='existingWikiWord' href='/nlab/show/Coxeter+group'>Coxeter group</a> <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>443</mn></msub></mrow><annotation encoding='application/x-tex'>C_{443}</annotation></semantics></math> is given by a <a class='existingWikiWord' href='/nlab/show/group+presentation'>group presentation</a> with 12 generators (represented by the vertices) of order <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math> (so 12 relators of the form <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>x^2 = 1</annotation></semantics></math>), with a relation <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mi>y</mi><msup><mo stretchy='false'>)</mo> <mn>3</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>(x y)^3 = 1</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x, y</annotation></semantics></math> are adjacent vertices (so 11 relators, one for each edge), and <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mi>y</mi><mo>=</mo><mi>y</mi><mi>x</mi></mrow><annotation encoding='application/x-tex'>x y = y x</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x, y</annotation></semantics></math> are non-adjacent (55 more relators). This Coxeter group (12 generators, 78 relators) is infinite, but by modding out by another strange ‘spider’ relator</p>
2386 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>c</mi> <mn>1</mn></msub><mi>a</mi><msub><mi>b</mi> <mn>2</mn></msub><msub><mi>c</mi> <mn>2</mn></msub><mi>a</mi><msub><mi>b</mi> <mn>3</mn></msub><msub><mi>c</mi> <mn>3</mn></msub><msup><mo stretchy='false'>)</mo> <mn>10</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>(a b_1 c_1 a b_2 c_2 a b_3 c_3)^{10} = 1</annotation></semantics></math></div>
2387 <p>the resulting quotient <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> turns out to be a <a class='existingWikiWord' href='/nlab/show/finite+group'>finite group</a>. Here <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> is the central vertex of valence <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>b_1, c_1</annotation></semantics></math> are on an arm of length <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>4</mn></mrow><annotation encoding='application/x-tex'>4</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>b_1</annotation></semantics></math> adjacent to <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>≠</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>c_1 \neq a</annotation></semantics></math> adjacent to <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>b_1</annotation></semantics></math>; similarly for <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>b_2, c_2</annotation></semantics></math> on the other arm of length <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>4</mn></mrow><annotation encoding='application/x-tex'>4</annotation></semantics></math>, and for <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>b_3, c_3</annotation></semantics></math> on the arm of length <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>. See <a href='http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html'>here</a> if this is not clear.</p>
2388
2389 <p>It turns out that <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/center'>center</a> <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> of order <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>, and the Monster <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is the quotient, i.e. the indicated term in the exact sequence</p>
2390 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>C</mi><mo>→</mo><mi>N</mi><mo>→</mo><mi>M</mi><mo>→</mo><mn>1</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'>1 \to C \to N \to M \to 1.</annotation></semantics></math></div>
2391 <p>This implicitly describes the Monster in terms of 12 generators and 80 relators.</p>
2392
2393 <p>Such “<math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>-group” presentations (Coxeter group based on a similar <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>-diagram, modulo a spider relation) are linked to a number of finite simple group constructions, the most famous of which is perhaps <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mn>555</mn></msub></mrow><annotation encoding='application/x-tex'>Y_{555}</annotation></semantics></math> which is a presentation of the “Bimonster” (the <a class='existingWikiWord' href='/nlab/show/wreath+product'>wreath product</a> of the Monster with <math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math>). See <a href='#Iv'>Ivanov</a> for a general description of these. The presentation of the Monster given above was established in <a href='#Iv2'>Ivanov2</a>.</p>
2394
2395 <h3 id='ViaAutomorphisms'>Via automorphisms of a super vertex operator algebra</h3>
2396
2397 <p>There is a <a class='existingWikiWord' href='/nlab/show/super+vertex+operator+algebra'>super vertex operator algebra</a>, the <a class='existingWikiWord' href='/nlab/show/Monster+vertex+operator+algebra'>Monster vertex operator algebra</a>, whose <a class='existingWikiWord' href='/nlab/show/automorphism'>group of</a> of <a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex+operator+algebra'>automorphisms of a VOA</a> is the <a class='existingWikiWord' href='/nlab/show/Monster+group'>monster group</a>.</p>
2398
2399 <p>(<a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a>, <a href='#GriessLam11'>Griess-Lam 11</a>)</p>
2400
2401 <h2 id='related_concepts_2'>Related concepts</h2>
2402
2403 <ul>
2404 <li>
2405 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine</a>,</p>
2406 </li>
2407
2408 <li>
2409 <p><a class='existingWikiWord' href='/nlab/show/Monster+vertex+operator+algebra'>Monster vertex algebra</a></p>
2410 </li>
2411
2412 <li>
2413 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu group</a>, <a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathieu moonshine</a></p>
2414 </li>
2415 </ul>
2416
2417 <h2 id='references'>References</h2>
2418
2419 <ul>
2420 <li>
2421 <p><a href='http://mathoverflow.net/users/39521/adam-p-goucher'>Adam P. Goucher</a>, <em>Presentation of the Monster Group</em>, (<a href='http://mathoverflow.net/q/142216'>MO comment 2013-09-15</a>)</p>
2422 </li>
2423
2424 <li id='Iv'>
2425 <p>Alexander Ivanov, <em>Y-groups via transitive extension</em>, Journal of Algebra, Volume 218, Issue 2 (August 15, 1999), 412–435. (<a href='http://www.sciencedirect.com/science/article/pii/S0021869399978821'>web</a>)</p>
2426 </li>
2427
2428 <li id='Iv2'>
2429 <p>A. A. Ivanov, <em>Constructing the Monster via its Y-presentation</em>, in Combinatorics, Paul Erdős is Eighty, Bolyai Society Mathematical Studies, Vol. 1 (1993), 253-270.</p>
2430 </li>
2431
2432 <li id='FrenkelLepowskiMeurman89'>
2433 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenkel</a>, <a class='existingWikiWord' href='/nlab/show/James+Lepowsky'>James Lepowsky</a>, Arne Meurman, <em>Vertex operator algebras and the monster</em>, Pure and Applied Mathematics <strong>134</strong>, Academic Press, New York 1998. liv+508 pp. <a href='http://www.ams.org/mathscinet-getitem?mr=996026'>MR0996026</a></p>
2434 </li>
2435
2436 <li id='GriessLam11'>
2437 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Griess</a> Jr., Ching Hung Lam, <em>A new existence proof of the Monster by VOA theory</em> (<a href='https://arxiv.org/abs/1103.1414'>arXiv:1103.1414</a>)</p>
2438 </li>
2439
2440 <li>
2441 <p><a class='existingWikiWord' href='/nlab/show/Andr%C3%A9+Henriques'>Andre Henriques</a>, <em><a href='http://mathoverflow.net/questions/69222/h4-of-the-monster'><math class='maruku-mathml' display='inline' id='mathml_b35f69f4bff39d523891a433ebff1bc914c95b59_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>H^4</annotation></semantics></math> of the monster</a></em></p>
2442 </li>
2443 </ul>
2444
2445 <p>Possible relation to <a class='existingWikiWord' href='/nlab/show/bosonic+M-theory'>bosonic M-theory</a>:</p>
2446
2447 <ul>
2448 <li><a class='existingWikiWord' href='/nlab/show/Alessio+Marrani'>Alessio Marrani</a>, <a class='existingWikiWord' href='/nlab/show/Michael+Rios'>Michael Rios</a>, <a class='existingWikiWord' href='/nlab/show/David+Chester'>David Chester</a>, <em>Monstrous M-theory</em> (<a href='https://arxiv.org/abs/2008.06742'>arXiv:2008.06742</a>)</li>
2449 </ul>
2450 <div style='float: right; margin: 0 20px 10px 20px;'><img alt='The Monster' src='http://t0.gstatic.com/images?q=tbn:nJNML0QhNiejuM:http://open.salon.com/files/cookie-monster3-7769871237963363.jpg ' width='80'/></div>
2451 <p>
2452 </p>
2453
2454 <p>
2455 </p>
2456
2457 <p>
2458 </p> </div>
2459 </content>
2460 </entry>
2461 <entry>
2462 <title type="html">Moonshine</title>
2463 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Moonshine"/>
2464 <updated>2021-07-01T21:06:02Z</updated>
2465 <published>2010-05-18T20:54:08Z</published>
2466 <id>tag:ncatlab.org,2010-05-18:nLab,Moonshine</id>
2467 <author>
2468 <name>Urs Schreiber</name>
2469 </author>
2470 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Moonshine">
2471 <div xmlns="http://www.w3.org/1999/xhtml">
2472 <div class='rightHandSide'>
2473 <div class='toc clickDown' tabindex='0'>
2474 <h3 id='context'>Context</h3>
2475
2476 <h4 id='exceptional_structures'>Exceptional structures</h4>
2477
2478 <div class='hide'>
2479 <p><strong><a class='existingWikiWord' href='/nlab/show/exceptional+structure'>exceptional structures</a></strong>, <a class='existingWikiWord' href='/nlab/show/exceptional+isomorphism'>exceptional isomorphisms</a></p>
2480
2481 <h2 id='examples'>Examples</h2>
2482
2483 <ul>
2484 <li>
2485 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>exceptional finite groups</a></p>
2486
2487 <ul>
2488 <li>
2489 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>monster group</a></p>
2490 </li>
2491
2492 <li>
2493 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu group</a>,</p>
2494 </li>
2495
2496 <li>
2497 <p><a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway group</a></p>
2498 </li>
2499 </ul>
2500 </li>
2501
2502 <li>
2503 <p>exceptional <a class='existingWikiWord' href='/nlab/show/finite+rotation+group'>finite rotation groups</a>:</p>
2504
2505 <ul>
2506 <li>
2507 <p><a class='existingWikiWord' href='/nlab/show/tetrahedral+group'>tetrahedral group</a></p>
2508 </li>
2509
2510 <li>
2511 <p><a class='existingWikiWord' href='/nlab/show/octahedral+group'>octahedral group</a></p>
2512 </li>
2513
2514 <li>
2515 <p><a class='existingWikiWord' href='/nlab/show/icosahedral+group'>icosahedral group</a></p>
2516 </li>
2517 </ul>
2518 </li>
2519
2520 <li>
2521 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+group'>exceptional Lie groups</a></p>
2522
2523 <ul>
2524 <li>
2525 <p><a class='existingWikiWord' href='/nlab/show/G2'>G2</a></p>
2526 </li>
2527
2528 <li>
2529 <p><a class='existingWikiWord' href='/nlab/show/F4'>F4</a></p>
2530 </li>
2531
2532 <li>
2533 <p><a class='existingWikiWord' href='/nlab/show/E6'>E6</a>, <a class='existingWikiWord' href='/nlab/show/E7'>E7</a>, <a class='existingWikiWord' href='/nlab/show/E8'>E8</a></p>
2534 </li>
2535 </ul>
2536
2537 <p>and <a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moody groups</a>:</p>
2538
2539 <ul>
2540 <li><a class='existingWikiWord' href='/nlab/show/E9'>E9</a>, <a class='existingWikiWord' href='/nlab/show/E10'>E10</a>, <a class='existingWikiWord' href='/nlab/show/E11'>E11</a>, …</li>
2541 </ul>
2542 </li>
2543
2544 <li>
2545 <p><a class='existingWikiWord' href='/nlab/show/Dwyer-Wilkerson+H-space'>Dwyer-Wilkerson H-space</a></p>
2546 </li>
2547
2548 <li>
2549 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+algebra'>exceptional Lie algebras</a></p>
2550 </li>
2551
2552 <li>
2553 <p><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>exceptional Jordan algebra</a></p>
2554
2555 <ul>
2556 <li><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>Albert algebra</a></li>
2557 </ul>
2558 </li>
2559
2560 <li>
2561 <p>exceptional <a class='existingWikiWord' href='/nlab/show/Jordan+superalgebra'>Jordan superalgebra</a>, <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mn>10</mn></msub></mrow><annotation encoding='application/x-tex'>K_10</annotation></semantics></math></p>
2562 </li>
2563
2564 <li>
2565 <p><a class='existingWikiWord' href='/nlab/show/Leech+lattice'>Leech lattice</a></p>
2566 </li>
2567
2568 <li>
2569 <p><a class='existingWikiWord' href='/nlab/show/Cayley+plane'>Cayley plane</a></p>
2570 </li>
2571 </ul>
2572
2573 <h2 id='interrelations'>Interrelations</h2>
2574
2575 <ul>
2576 <li>
2577 <p><a class='existingWikiWord' href='/nlab/show/division+algebra+and+supersymmetry'>supersymmetry and division algebras</a></p>
2578 </li>
2579
2580 <li>
2581 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+magic+square'>Freudenthal magic square</a></p>
2582 </li>
2583
2584 <li>
2585 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>moonshine</a></p>
2586
2587 <ul>
2588 <li>
2589 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathieu moonshine</a></p>
2590 </li>
2591
2592 <li>
2593 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral moonshine</a></p>
2594 </li>
2595
2596 <li>
2597 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O&#39;Nan moonshine</a></p>
2598 </li>
2599 </ul>
2600 </li>
2601 </ul>
2602
2603 <h2 id='applications'>Applications</h2>
2604
2605 <ul>
2606 <li>
2607 <p><a class='existingWikiWord' href='/nlab/show/exceptional+geometry'>exceptional geometry</a>, <a class='existingWikiWord' href='/nlab/show/exceptional+generalized+geometry'>exceptional generalized geometry</a>,</p>
2608 </li>
2609
2610 <li>
2611 <p><a class='existingWikiWord' href='/nlab/show/exceptional+field+theory'>exceptional field theory</a></p>
2612 </li>
2613 </ul>
2614
2615 <h2 id='philosophy'>Philosophy</h2>
2616
2617 <ul>
2618 <li><a class='existingWikiWord' href='/nlab/show/universal+exceptionalism'>universal exceptionalism</a></li>
2619 </ul>
2620 </div>
2621 </div>
2622 </div>
2623
2624 <h1 id='contents'>Contents</h1>
2625 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#AutomorphismGroupsOfVertexOperatorAlgebras'>Automorphism groups of vertex operator algebras</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#historical_references'>Historical References</a></li><li><a href='#FurtherDevelomentsReferences'>Further developments</a></li><li><a href='#realization_in_superstring_theory'>Realization in superstring theory</a></li></ul></li></ul></div>
2626 <h2 id='idea'>Idea</h2>
2627
2628 <p>Moonshine usually refers to the mysterious connections between the <a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster simple group</a> and the modular function <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/j-invariant'>j-invariant</a>. There were a bunch of <a class='existingWikiWord' href='/nlab/show/conjecture'>conjectures</a> about this connection that were proved by <a class='existingWikiWord' href='/nlab/show/Richard+Borcherds'>Richard Borcherds</a>, en passant mentioning the existence of the <a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine vertex algebra</a> (constructed then later in <a href='#FrenkelLepowskiMeurman89'>FLM 89</a>). Nowadays there is also Moonshine for other simple groups, by the work of J. Duncan. Eventually there should be an entry for the general moonshine phenomenon.</p>
2629
2630 <p>The whole idea of moonshine began with <a class='existingWikiWord' href='/nlab/show/John+McKay'>John McKay</a>’s observation that the <a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster group</a>’s first nontrivial <a class='existingWikiWord' href='/nlab/show/irreducible+representation'>irreducible representation</a> has <a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a> 196883, and the <a class='existingWikiWord' href='/nlab/show/j-invariant'>j-invariant</a> <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>j(\tau)</annotation></semantics></math> has the <a class='existingWikiWord' href='/nlab/show/Fourier+transform'>Fourier series</a> expansion</p>
2631 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>τ</mi><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>744</mn><mo>+</mo><mn>196884</mn><mi>q</mi><mo>+</mo><mn>21493760</mn><msup><mi>q</mi> <mn>2</mn></msup><mo>+</mo><mi>…</mi></mrow><annotation encoding='application/x-tex'>
2632 j(\tau) = q^{-1} + 744 + 196884q + 21493760q^{2} + \dots
2633
2634 </annotation></semantics></math></div>
2635 <p>where <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi><mo>=</mo><mi>exp</mi><mo stretchy='false'>(</mo><mi>i</mi><mn>2</mn><mi>π</mi><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>q=\exp(i2\pi\tau)</annotation></semantics></math>, and famously 196883+1=196884. Thompson observed in (1979) that the other coefficients are obtained from the dimensions of Monster’s irreducible representations.</p>
2636
2637 <p>But the monster was merely <em>conjectured</em> to exist until Griess (1982) explicitly constructed it. The construction is horribly complicated (take the sum of three irreducible representations for the <a class='existingWikiWord' href='/nlab/show/centralizer'>centralizer</a> of an <a class='existingWikiWord' href='/nlab/show/involution'>involution</a> of…).</p>
2638
2639 <p><a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a> constructed an infinite-dimensional <a class='existingWikiWord' href='/nlab/show/module'>module</a> for the <a class='existingWikiWord' href='/nlab/show/Monster+vertex+operator+algebra'>Monster vertex algebra</a>. This is by a generalized <a class='existingWikiWord' href='/nlab/show/Kac-Moody+algebra'>Kac-Moody algebra</a> via <a class='existingWikiWord' href='/nlab/show/bosonic+string+theory'>bosonic string theory</a> and the <a class='existingWikiWord' href='/nlab/show/Goddard-Thorn+theorem'>Goddard-Thorn &quot;No Ghost&quot; theorem</a>. The <a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster group</a> acts naturally on this “Moonshine Module” (denoted by <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>♮</mo></mrow><annotation encoding='application/x-tex'>V\natural</annotation></semantics></math>).</p>
2640
2641 <p>To cut the story short, we end up getting from the Monster group to a module it acts on which is related to “modular stuff” (namely, the modular <a class='existingWikiWord' href='/nlab/show/j-invariant'>j-invariant</a>). The idea <a class='existingWikiWord' href='/nlab/show/Terry+Gannon'>Terry Gannon</a> pitches is that Moonshine is a generalization of this association, it’s a sort of “mapping” from “Algebraic gadgets” to “Modular stuff”.</p>
2642
2643 <h2 id='AutomorphismGroupsOfVertexOperatorAlgebras'>Automorphism groups of vertex operator algebras</h2>
2644
2645 <p>Realizations of <a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>sporadic finite simple groups</a> as <span class='newWikiWord'>automorphism groups of vertex operator algebras<a href='/nlab/new/automorphism+groups+of+vertex+operator+algebras'>?</a></span> in <a class='existingWikiWord' href='/nlab/show/heterotic+string+theory'>heterotic string theory</a> and <a class='existingWikiWord' href='/nlab/show/type+II+string+theory'>type II string theory</a> (mostly on <a class='existingWikiWord' href='/nlab/show/K3+surface'>K3-surfaces</a>, see <a class='existingWikiWord' href='/nlab/show/duality+between+heterotic+and+type+II+string+theory'>HET - II duality</a>):</p>
2646
2647 <ul>
2648 <li>
2649 <p>The <a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway group</a> <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Co</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>Co_{0}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/automorphism'>group of</a> <a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex+operator+algebra'>automorphisms of a super VOA</a> of the unique chiral <a class='existingWikiWord' href='/nlab/show/number+of+supersymmetries'>N=1</a> <a class='existingWikiWord' href='/nlab/show/super+vertex+operator+algebra'>super vertex operator algebra</a> of <a class='existingWikiWord' href='/nlab/show/central+charge'>central charge</a> <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>=</mo><mn>12</mn></mrow><annotation encoding='application/x-tex'>c = 12</annotation></semantics></math> without fields of <a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>conformal weight</a> <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>1/2</annotation></semantics></math></p>
2650
2651 <p>(<a href='#Duncan05'>Duncan 05</a>, see also <a href='#PaquettePerssonVolpato17'>Paquette-Persson-Volpato 17, p. 9</a>)</p>
2652 </li>
2653
2654 <li>
2655 <p>similarly, there is a super VOA, the <em><a class='existingWikiWord' href='/nlab/show/Monster+vertex+operator+algebra'>Monster vertex operator algebra</a></em>, whose <a class='existingWikiWord' href='/nlab/show/automorphism'>group of</a> of <a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex+operator+algebra'>automorphisms of a VOA</a> is the <a class='existingWikiWord' href='/nlab/show/Monster+group'>monster group</a></p>
2656
2657 <p>(<a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a>, <a href='#GriessLam11'>Griess-Lam 11</a>)</p>
2658 </li>
2659 </ul>
2660
2661 <h2 id='related_concepts'>Related concepts</h2>
2662
2663 <ul>
2664 <li>
2665 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster</a></p>
2666 </li>
2667
2668 <li>
2669 <p>moonshine</p>
2670
2671 <ul>
2672 <li>
2673 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathieu moonshine</a></p>
2674 </li>
2675
2676 <li>
2677 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral moonshine</a></p>
2678 </li>
2679
2680 <li>
2681 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O&#39;Nan moonshine</a></p>
2682 </li>
2683 </ul>
2684 </li>
2685
2686 <li>
2687 <p><a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex+operator+algebra'>automorphism of a vertex operator algebra</a></p>
2688 </li>
2689 </ul>
2690
2691 <h2 id='references'>References</h2>
2692
2693 <h3 id='general'>General</h3>
2694
2695 <ul>
2696 <li>
2697 <p><a class='existingWikiWord' href='/nlab/show/Richard+Borcherds'>Richard Borcherds</a>, <em>What is Moonshine?, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).</em>Doc. Math._ 1998, Extra Vol. I, 607–615 (electronic). <a href='http://www.ams.org/mathscinet-getitem?mr=1660657'>MR1660657</a> <a href='http://arxiv.org/abs/math/9809110'>arXiv:math/9809110v1</a> [math.QA]</p>
2698 </li>
2699
2700 <li>
2701 <p>John F. R. Duncan, Michael J. Griffin, Ken Ono, <em>Moonshine</em> (<a href='http://arxiv.org/abs/1411.6571'>arXiv:1411.6571</a>)</p>
2702 </li>
2703
2704 <li id='GriessLam11'>
2705 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Griess</a> Jr., Ching Hung Lam, <em>A new existence proof of the Monster by VOA theory</em> (<a href='https://arxiv.org/abs/1103.1414'>arXiv:1103.1414</a>)</p>
2706 </li>
2707
2708 <li id='FrenkelLepowskiMeurman89'>
2709 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenkel</a>, <a class='existingWikiWord' href='/nlab/show/James+Lepowsky'>James Lepowsky</a>, Arne Meurman, <em>Vertex operator algebras and the monster</em>, Pure and Applied Mathematics <strong>134</strong>, Academic Press, New York 1989. liv+508 pp. <a href='http://www.ams.org/mathscinet-getitem?mr=996026'>MR0996026</a></p>
2710 </li>
2711
2712 <li>
2713 <p><a class='existingWikiWord' href='/nlab/show/Terry+Gannon'>Terry Gannon</a>, <em>Monstrous moonshine: the first twenty-five years</em>, <em>Bull. London Math. Soc.</em> <strong>38</strong> (2006), no. 1, 1–33. <a href='http://www.ams.org/mathscinet-getitem?mr=2201600'>MR2201600</a> <a href='http://arxiv.org/abs/math/0402345'>arXiv:math/0402345</a> [math.QA]</p>
2714 </li>
2715
2716 <li>
2717 <p><a class='existingWikiWord' href='/nlab/show/Terry+Gannon'>Terry Gannon</a>, <em>Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics</em>, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Massachusetts 2006. <a href='http://www.ams.org/mathscinet-getitem?mr=2257727'>MR2257727</a></p>
2718 </li>
2719
2720 <li>
2721 <p>Koichiro Harada, <em>“Moonshine” of finite groups</em>. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2010. viii+76 pp. <a href='http://www.ams.org/mathscinet-getitem?mr=2722318'>MR2722318</a></p>
2722 </li>
2723
2724 <li>
2725 <p>Griess, Robert L., Jr.; Lam, Ching Hung <em>A moonshine path from E8 to the Monster</em>, J. Pure Appl. Algebra_ 215 (2011), no. 5, 927–948 <a href='http://www.ams.org/mathscinet-getitem?mr=2747229'>MR2747229</a> <a href='http://arxiv.org/abs/0910.2057v2'>arXiv:0910.2057v2</a> [math.GR]</p>
2726 </li>
2727
2728 <li>
2729 <p>Jae-Hyun Yang “Kac-Moody algebras, the Monstrous Moonshine, Jacobi forms and infinite products.” <em>Number theory, geometry and related topics</em> (Iksan City, 1995), 13–82, Pyungsan Inst. Math. Sci., Seoul, 1996. <a href='http://www.ams.org/mathscinet-getitem?mr=1404967'>MR1404967</a> <a href='http://arxiv.org/abs/math/0612474'>arXiv:math/0612474v2</a> [math.NT]</p>
2730 </li>
2731
2732 <li>
2733 <p>Vassilis Anagiannis, <a class='existingWikiWord' href='/nlab/show/Miranda+Cheng'>Miranda Cheng</a>, <em>TASI Lectures on Moonshine</em> (<a href='https://arxiv.org/abs/1807.00723'>arXiv:1807.00723</a>)</p>
2734 </li>
2735 </ul>
2736
2737 <h3 id='historical_references'>Historical References</h3>
2738
2739 <ul>
2740 <li>
2741 <p><a class='existingWikiWord' href='/nlab/show/John+Horton+Conway'>John Conway</a> and Simon Norton, “Monstrous moonshine.” <em>Bull. London Math. Soc.</em> <strong>11</strong> (1979), no. 3, 308–339; <a href='http://www.ams.org/mathscinet-getitem?mr=554399'>MR0554399</a> (81j:20028)</p>
2742 </li>
2743
2744 <li id='FrenkelLepowskiMeurman89'>
2745 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenkel</a>, <a class='existingWikiWord' href='/nlab/show/James+Lepowsky'>James Lepowsky</a>, Arne Meurman, “A natural representation of the Fischer-Griess Monster with the modular function <math class='maruku-mathml' display='inline' id='mathml_5fa857844ec088dd2601ac2b4c1e27b3d0f3ef74_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> as character.” <em>Proc. Nat. Acad. Sci. U.S.A.</em> <strong>81</strong> (1984), no. 10, Phys. Sci., 3256–3260. <a href='http://www.ams.org/mathscinet-getitem?mr=747596'>MR0747596</a> (85e:20018)</p>
2746 </li>
2747
2748 <li>
2749 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Griess</a>, “The friendly giant.” <em>Invent. Math.</em> <strong>69</strong> (1982), no. 1, 1–102. <a href='http://www.ams.org/mathscinet-getitem?mr=671653'>MR671653</a> (84m:20024)</p>
2750 </li>
2751
2752 <li>
2753 <p>John G. Thompson, “Some numerology between the Fischer-Griess Monster and the elliptic modular function.” <em>Bull. London Math. Soc.</em> <strong>11</strong> (1979), no. 3, 352–353. <a href='http://www.ams.org/mathscinet-getitem?mr=554402'>MR0554402</a> (81j:20030)</p>
2754 </li>
2755 </ul>
2756
2757 <h3 id='FurtherDevelomentsReferences'>Further developments</h3>
2758
2759 <ul>
2760 <li>
2761 <p><a class='existingWikiWord' href='/nlab/show/Miranda+Cheng'>Miranda Cheng</a>, John F. R. Duncan, Jeffrey A. Harvey, <em>Umbral Moonshine</em> (<a href='http://arxiv.org/abs/1204.2779'>arXiv:1204.2779</a>)</p>
2762 </li>
2763
2764 <li>
2765 <p>John F. R. Duncan, Michael J. Griffin and Ken Ono, <em>Proof of the Umbral Moonshine Conjecture</em> (<a href='http://arxiv.org/abs/1503.01472'>arXiv:1503.01472</a>)</p>
2766 </li>
2767
2768 <li>
2769 <p><a class='existingWikiWord' href='/nlab/show/Scott+Carnahan'>Scott Carnahan</a>, <em>Monstrous Moonshine over Z?</em> (<a href='https://arxiv.org/abs/1804.04161'>arXiv:1804.04161</a>)</p>
2770 </li>
2771 </ul>
2772
2773 <h3 id='realization_in_superstring_theory'>Realization in superstring theory</h3>
2774
2775 <p>Discussion of possible realizations in <a class='existingWikiWord' href='/nlab/show/string+theory'>superstring theory</a> (specifically <a class='existingWikiWord' href='/nlab/show/heterotic+string+theory'>heterotic string theory</a> and <a class='existingWikiWord' href='/nlab/show/type+II+string+theory'>type II string theory</a> in <a class='existingWikiWord' href='/nlab/show/K3+surface'>K3-surfaces</a>, see <a class='existingWikiWord' href='/nlab/show/duality+between+heterotic+and+type+II+string+theory'>HET - II</a>) via <a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex+operator+algebra'>automorphisms of super vertex operator algebras</a>:</p>
2776
2777 <ul>
2778 <li>
2779 <p>S. Chaudhuri, D.A. Lowe, <em>Monstrous String-String Duality</em>, Nucl. Phys. B469 : 21-36, 1996 (<a href='https://arxiv.org/abs/hep-th/9512226'>arXiv:hep-th/9512226</a>)</p>
2780 </li>
2781
2782 <li id='Duncan05'>
2783 <p>John F. Duncan, <em>Super-moonshine for Conway’s largest sporadic group</em> (<a href='https://arxiv.org/abs/math/0502267'>arXiv:math/0502267</a>)</p>
2784 </li>
2785
2786 <li id='PaquettePerssonVolpato16'>
2787 <p><a class='existingWikiWord' href='/nlab/show/Natalie+Paquette'>Natalie Paquette</a>, Daniel Persson, Roberto Volpato, <em>Monstrous BPS-Algebras and the Superstring Origin of Moonshine</em> (<a href='http://arxiv.org/abs/1601.05412'>arXiv:1601.05412</a>)</p>
2788 </li>
2789
2790 <li id='KachruPaquetteVolpato16'>
2791 <p><a class='existingWikiWord' href='/nlab/show/Shamit+Kachru'>Shamit Kachru</a>, <a class='existingWikiWord' href='/nlab/show/Natalie+Paquette'>Natalie Paquette</a>, Roberto Volpato, <em>3D String Theory and Umbral Moonshine</em> (<a href='http://arxiv.org/abs/1603.07330'>arXiv:1603.07330</a>)</p>
2792 </li>
2793
2794 <li id='PaquettePerssonVolpato17'>
2795 <p><a class='existingWikiWord' href='/nlab/show/Natalie+Paquette'>Natalie Paquette</a>, Daniel Persson, Roberto Volpato, <em>BPS Algebras, Genus Zero, and the Heterotic Monster</em> (<a href='https://arxiv.org/abs/1701.05169'>arXiv:1701.05169</a>)</p>
2796 </li>
2797
2798 <li>
2799 <p><a class='existingWikiWord' href='/nlab/show/Shamit+Kachru'>Shamit Kachru</a>, Arnav Tripathy, <em>The hidden symmetry of the heterotic string</em> (<a href='https://arxiv.org/abs/1702.02572'>arXiv:1702.02572</a>)</p>
2800 </li>
2801 </ul>
2802
2803 <p>Specifically in relation to <a class='existingWikiWord' href='/nlab/show/Kaluza-Klein+mechanism'>KK-compactifications</a> of <a class='existingWikiWord' href='/nlab/show/string+theory'>string theory</a> on <a class='existingWikiWord' href='/nlab/show/K3+surface'>K3-surfaces</a> (<a class='existingWikiWord' href='/nlab/show/duality+between+heterotic+and+type+II+string+theory'>duality between heterotic and type II string theory</a>)</p>
2804
2805 <ul>
2806 <li id='ChengHarrisonVolpatoZimet16'><a class='existingWikiWord' href='/nlab/show/Miranda+Cheng'>Miranda Cheng</a>, Sarah M. Harrison, Roberto Volpato, Max Zimet, <em>K3 String Theory, Lattices and Moonshine</em> (<a href='https://arxiv.org/abs/1612.04404'>arXiv:1612.04404</a>, <a href='https://doi.org/10.1007/s40687-018-0150-4'>doi:10.1007/s40687-018-0150-4</a>)</li>
2807 </ul>
2808
2809 <p>Possible relation to <a class='existingWikiWord' href='/nlab/show/bosonic+M-theory'>bosonic M-theory</a>:</p>
2810
2811 <ul>
2812 <li><a class='existingWikiWord' href='/nlab/show/Alessio+Marrani'>Alessio Marrani</a>, <a class='existingWikiWord' href='/nlab/show/Michael+Rios'>Michael Rios</a>, <a class='existingWikiWord' href='/nlab/show/David+Chester'>David Chester</a>, <em>Monstrous M-theory</em> (<a href='https://arxiv.org/abs/2008.06742'>arXiv:2008.06742</a>)</li>
2813 </ul>
2814
2815 <p>
2816
2817 </p>
2818
2819 <p>
2820
2821 </p>
2822
2823 <p>
2824
2825 </p> </div>
2826 </content>
2827 </entry>
2828 <entry>
2829 <title type="html">geometric realization of categories</title>
2830 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/geometric+realization+of+categories"/>
2831 <updated>2021-07-01T17:07:36Z</updated>
2832 <published>2011-05-30T18:19:35Z</published>
2833 <id>tag:ncatlab.org,2011-05-30:nLab,geometric+realization+of+categories</id>
2834 <author>
2835 <name>Dmitri Pavlov</name>
2836 </author>
2837 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/geometric+realization+of+categories">
2838 <div xmlns="http://www.w3.org/1999/xhtml">
2839 <div class='rightHandSide'>
2840 <div class='toc clickDown' tabindex='0'>
2841 <h3 id='context'>Context</h3>
2842
2843 <h4 id='homotopy_theory'>Homotopy theory</h4>
2844
2845 <div class='hide'>
2846 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a></strong></p>
2847
2848 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/directed+homotopy+theory'>directed</a>…</p>
2849
2850 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/localic+homotopy+theory'>localic</a>, …</p>
2851
2852 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong></p>
2853
2854 <p><strong>Introductions</strong></p>
2855
2856 <ul>
2857 <li>
2858 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p>
2859 </li>
2860
2861 <li>
2862 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p>
2863 </li>
2864
2865 <li>
2866 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p>
2867 </li>
2868 </ul>
2869
2870 <p><strong>Definitions</strong></p>
2871
2872 <ul>
2873 <li>
2874 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/higher+homotopy'>higher homotopy</a></p>
2875 </li>
2876
2877 <li>
2878 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p>
2879 </li>
2880
2881 <li>
2882 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/spherical+object'>spherical object and Pi(A)-algebra</a></p>
2883 </li>
2884
2885 <li>
2886 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p>
2887
2888 <ul>
2889 <li>
2890 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></p>
2891
2892 <ul>
2893 <li>
2894 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p>
2895 </li>
2896
2897 <li>
2898 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/cofibration+category'>cofibration category</a></p>
2899 </li>
2900
2901 <li>
2902 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Waldhausen category</a></p>
2903 </li>
2904 </ul>
2905 </li>
2906
2907 <li>
2908 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></p>
2909
2910 <ul>
2911 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a></li>
2912 </ul>
2913 </li>
2914 </ul>
2915 </li>
2916
2917 <li>
2918 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
2919
2920 <ul>
2921 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li>
2922 </ul>
2923 </li>
2924 </ul>
2925
2926 <p><strong>Paths and cylinders</strong></p>
2927
2928 <ul>
2929 <li>
2930 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a></p>
2931
2932 <ul>
2933 <li>
2934 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a></p>
2935 </li>
2936
2937 <li>
2938 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a></p>
2939 </li>
2940 </ul>
2941 </li>
2942
2943 <li>
2944 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
2945
2946 <ul>
2947 <li>
2948 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path object</a></p>
2949 </li>
2950
2951 <li>
2952 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
2953 </li>
2954
2955 <li>
2956 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle'>universal bundle</a></p>
2957 </li>
2958 </ul>
2959 </li>
2960
2961 <li>
2962 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a></p>
2963
2964 <ul>
2965 <li>
2966 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometric+homotopies'>homotopy localization</a></p>
2967 </li>
2968
2969 <li>
2970 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+object'>infinitesimal interval object</a></p>
2971 </li>
2972 </ul>
2973 </li>
2974 </ul>
2975
2976 <p><strong>Homotopy groups</strong></p>
2977
2978 <ul>
2979 <li>
2980 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
2981
2982 <ul>
2983 <li>
2984 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a></p>
2985
2986 <ul>
2987 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+topos'>fundamental group of a topos</a></li>
2988 </ul>
2989 </li>
2990
2991 <li>
2992 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p>
2993 </li>
2994
2995 <li>
2996 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p>
2997 </li>
2998
2999 <li>
3000 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p>
3001 </li>
3002 </ul>
3003 </li>
3004
3005 <li>
3006 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p>
3007
3008 <ul>
3009 <li>
3010 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a></p>
3011
3012 <ul>
3013 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path groupoid</a></li>
3014 </ul>
3015 </li>
3016
3017 <li>
3018 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p>
3019 </li>
3020
3021 <li>
3022 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p>
3023 </li>
3024 </ul>
3025 </li>
3026
3027 <li>
3028 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p>
3029
3030 <ul>
3031 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental category</a></li>
3032 </ul>
3033 </li>
3034 </ul>
3035
3036 <p><strong>Basic facts</strong></p>
3037
3038 <ul>
3039 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li>
3040 </ul>
3041
3042 <p><strong>Theorems</strong></p>
3043
3044 <ul>
3045 <li>
3046 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
3047 </li>
3048
3049 <li>
3050 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
3051 </li>
3052
3053 <li>
3054 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p>
3055 </li>
3056
3057 <li>
3058 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p>
3059 </li>
3060
3061 <li>
3062 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
3063 </li>
3064
3065 <li>
3066 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead&#39;s theorem</a></p>
3067 </li>
3068
3069 <li>
3070 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewicz theorem</a></p>
3071 </li>
3072
3073 <li>
3074 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois theory</a></p>
3075 </li>
3076
3077 <li>
3078 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p>
3079 </li>
3080 </ul>
3081 </div>
3082
3083 <h4 id='category_theory'>Category theory</h4>
3084
3085 <div class='hide'>
3086 <p><strong><a class='existingWikiWord' href='/nlab/show/category+theory'>category theory</a></strong></p>
3087
3088 <h2 id='concepts'>Concepts</h2>
3089
3090 <ul>
3091 <li>
3092 <p><a class='existingWikiWord' href='/nlab/show/category'>category</a></p>
3093 </li>
3094
3095 <li>
3096 <p><a class='existingWikiWord' href='/nlab/show/functor'>functor</a></p>
3097 </li>
3098
3099 <li>
3100 <p><a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a></p>
3101 </li>
3102
3103 <li>
3104 <p><a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a></p>
3105 </li>
3106 </ul>
3107
3108 <h2 id='universal_constructions'>Universal constructions</h2>
3109
3110 <ul>
3111 <li>
3112 <p><a class='existingWikiWord' href='/nlab/show/universal+construction'>universal construction</a></p>
3113
3114 <ul>
3115 <li>
3116 <p><a class='existingWikiWord' href='/nlab/show/representable+functor'>representable functor</a></p>
3117 </li>
3118
3119 <li>
3120 <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functor</a></p>
3121 </li>
3122
3123 <li>
3124 <p><a class='existingWikiWord' href='/nlab/show/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/colimit'>colimit</a></p>
3125 </li>
3126
3127 <li>
3128 <p><a class='existingWikiWord' href='/nlab/show/weighted+limit'>weighted limit</a></p>
3129 </li>
3130
3131 <li>
3132 <p><a class='existingWikiWord' href='/nlab/show/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/end'>coend</a></p>
3133 </li>
3134
3135 <li>
3136 <p><a class='existingWikiWord' href='/nlab/show/Kan+extension'>Kan extension</a></p>
3137 </li>
3138 </ul>
3139 </li>
3140 </ul>
3141
3142 <h2 id='theorems'>Theorems</h2>
3143
3144 <ul>
3145 <li>
3146 <p><a class='existingWikiWord' href='/nlab/show/Yoneda+lemma'>Yoneda lemma</a></p>
3147 </li>
3148
3149 <li>
3150 <p><a class='existingWikiWord' href='/nlab/show/Isbell+duality'>Isbell duality</a></p>
3151 </li>
3152
3153 <li>
3154 <p><a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck construction</a></p>
3155 </li>
3156
3157 <li>
3158 <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor+theorem'>adjoint functor theorem</a></p>
3159 </li>
3160
3161 <li>
3162 <p><a class='existingWikiWord' href='/nlab/show/monadicity+theorem'>monadicity theorem</a></p>
3163 </li>
3164
3165 <li>
3166 <p><a class='existingWikiWord' href='/nlab/show/adjoint+lifting+theorem'>adjoint lifting theorem</a></p>
3167 </li>
3168
3169 <li>
3170 <p><a class='existingWikiWord' href='/nlab/show/Tannaka+duality'>Tannaka duality</a></p>
3171 </li>
3172
3173 <li>
3174 <p><a class='existingWikiWord' href='/nlab/show/Gabriel-Ulmer+duality'>Gabriel-Ulmer duality</a></p>
3175 </li>
3176
3177 <li>
3178 <p><a class='existingWikiWord' href='/nlab/show/small+object+argument'>small object argument</a></p>
3179 </li>
3180
3181 <li>
3182 <p><a class='existingWikiWord' href='/nlab/show/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p>
3183 </li>
3184
3185 <li>
3186 <p><a class='existingWikiWord' href='/nlab/show/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p>
3187 </li>
3188 </ul>
3189
3190 <h2 id='extensions'>Extensions</h2>
3191
3192 <ul>
3193 <li>
3194 <p><a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>sheaf and topos theory</a></p>
3195 </li>
3196
3197 <li>
3198 <p><a class='existingWikiWord' href='/nlab/show/enriched+category+theory'>enriched category theory</a></p>
3199 </li>
3200
3201 <li>
3202 <p><a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a></p>
3203 </li>
3204 </ul>
3205
3206 <h2 id='applications'>Applications</h2>
3207
3208 <ul>
3209 <li><a class='existingWikiWord' href='/nlab/show/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li>
3210 </ul>
3211 <div>
3212 <p>
3213 <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a>
3214 </p>
3215 </div></div>
3216 </div>
3217 </div>
3218
3219 <h1 id='contents'>Contents</h1>
3220 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#ThomasonModelStructure'>Thomason model structure</a></li><li><a href='#recognizing_weak_equivalences_quillens_theorem_a_and_b'>Recognizing weak equivalences: Quillen’s theorem A and B</a></li><li><a href='#natural_transformations_and_homotopies'>Natural transformations and homotopies</a></li><li><a href='#behaviour_under_homotopy_colimits'>Behaviour under homotopy colimits</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#quillens_theorems_a_and_b'>Quillen’s theorems A and B</a></li></ul></li></ul></div>
3221 <h2 id='idea'>Idea</h2>
3222
3223 <p>What is called <em>geometric realization of categories</em> is a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a> that sends <a class='existingWikiWord' href='/nlab/show/category'>categories</a> to <a class='existingWikiWord' href='/nlab/show/topological+space'>topological spaces</a>, namely the functor which first forms the <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial set</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(\mathcal{C})</annotation></semantics></math> that is the <a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a> of the category <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, and then forms the <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(\mathcal{C})\vert}</annotation></semantics></math> of this simplical set. Typically one is interested in this geometric realization up to <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a>.</p>
3224
3225 <p>By the <a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem the <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> of simplicial sets constitutes a (<a class='existingWikiWord' href='/nlab/show/Quillen+equivalence'>Quillen</a>)<a class='existingWikiWord' href='/nlab/show/equivalence+of+%28infinity%2C1%29-categories'>equivalence</a> between the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+simplicial+sets'>classical homotopy theory of simiplicial sets</a> and the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a>. This means that inasmuch as one is interested in geometric realization of categories up to weak homotopy equivalence, then the key part of the operation is in forming the simplicial nerve <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(\mathcal{C})</annotation></semantics></math> of a category, with the latter regarded as a model for an <a class='existingWikiWord' href='/nlab/show/infinity-groupoid'>∞-groupoid</a>. Indeed, equivalently one could consider the <a class='existingWikiWord' href='/nlab/show/Kan+fibrant+replacement'>Kan fibrant replacement</a> of the nerve <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(\mathcal{C})</annotation></semantics></math> (which still has the same geometric realization, up to weak homotopy equivalence).</p>
3226
3227 <p>Therefore an equivalent perspective on geometric realization of categories is that it universally turns a category into an <a class='existingWikiWord' href='/nlab/show/infinity-groupoid'>infinity-groupoid</a> by freely turning all its morphisms into <a class='existingWikiWord' href='/nlab/show/equivalence+in+an+%28infinity%2C1%29-category'>homotopy equivalences</a>.</p>
3228
3229 <p>Geometric realization of categories has various good properties:</p>
3230
3231 <p>It sends <a class='existingWikiWord' href='/nlab/show/equivalence+of+categories'>equivalences of categories</a> to <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalences</a> (corollary <a class='maruku-ref' href='#RealizationOfEquivalenceIsHomotopyEquivalence'>1</a> below). A more general sufficient criterion for the geometric realization of a functor is given by the seminal theorem known as <em>Quillen’s theorem A</em> (theorem <a class='maruku-ref' href='#QuillenTheoremA'>1</a> below.)</p>
3232
3233 <p>The existence of the <a class='existingWikiWord' href='/nlab/show/Thomason+model+structure'>Thomason model structure</a> (<a href='#ThomasonModelStructure'>below</a>) implies that every <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a> arises as the geometric realization of some category. In fact it already arises as the geometric realization of some <a class='existingWikiWord' href='/nlab/show/partial+order'>poset</a> (<a class='existingWikiWord' href='/nlab/show/%280%2C1%29-category'>(0,1)-category</a>).</p>
3234
3235 <h2 id='definition'>Definition</h2>
3236
3237 <p>Write</p>
3238 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo lspace='verythinmathspace'>:</mo><mi>Cat</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>
3239 N \colon Cat \to sSet
3240
3241 </annotation></semantics></math></div>
3242 <p>for the <a href='nerve#NerveOfACategory'>nerve functor</a> from <a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a> to <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>. Write</p>
3243 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mo>−</mo><mo stretchy='false'>|</mo></mrow><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>
3244 {\vert - \vert} : sSet \to Top
3245
3246 </annotation></semantics></math></div>
3247 <p>for the <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> of <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> from <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a> to <a class='existingWikiWord' href='/nlab/show/Top'>Top</a>.</p>
3248
3249 <p>The <em>geometric realization of categories</em> is the <a class='existingWikiWord' href='/nlab/show/composition'>composite</a> of these two operations:</p>
3250 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mo>−</mo><mo stretchy='false'>|</mo></mrow><mo>≔</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>Cat</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>
3251 {\vert - \vert} \coloneqq {\vert N(-)\vert} \;\colon\; Cat \to Top
3252
3253 </annotation></semantics></math></div>
3254 <h2 id='properties'>Properties</h2>
3255
3256 <h3 id='ThomasonModelStructure'>Thomason model structure</h3>
3257
3258 <p>There is a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> structure on <a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a> whose weak equivalences are those <a class='existingWikiWord' href='/nlab/show/functor'>functors</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>F \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> which under <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> become weak equivalences in the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a>, hence which become <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalences</a>. This is called the <em><a class='existingWikiWord' href='/nlab/show/Thomason+model+structure'>Thomason model structure</a></em>.</p>
3259
3260 <p>The existence of the Thomas model structure implies that every <a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a> arises as the geometric realization of some category, in fact already as the realization of some <a class='existingWikiWord' href='/nlab/show/partial+order'>poset</a>/<a class='existingWikiWord' href='/nlab/show/%280%2C1%29-category'>(0,1)-category</a>:</p>
3261
3262 <div class='num_defn' id='PosetOfSimplicesInNerveOfCategory'>
3263 <h6 id='definition_2'>Definition</h6>
3264
3265 <p>For <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/category'>category</a>, let <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>\nabla C</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/partial+order'>poset</a> of <a class='existingWikiWord' href='/nlab/show/simplex'>simplices</a> in the <a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mi>C</mi></mrow><annotation encoding='application/x-tex'>N C</annotation></semantics></math>, ordered by inclusion.</p>
3266 </div>
3267
3268 <div class='num_prop'>
3269 <h6 id='proposition'>Proposition</h6>
3270
3271 <p>For every category <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> the poset <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\nabla \mathcal{C}</annotation></semantics></math> from def. <a class='maruku-ref' href='#PosetOfSimplicesInNerveOfCategory'>1</a> has <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weakly homotopy equivalent</a> geometric realization</p>
3272 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mo>∇</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><msub><mo>≃</mo> <mi>wh</mi></msub><mrow><mo stretchy='false'>|</mo><mi>𝒞</mi><mo stretchy='false'>|</mo></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
3273 {\vert N(\nabla \mathcal{C}) \vert} \simeq_{wh} {\vert \mathcal{C} \vert}
3274 \,.
3275
3276 </annotation></semantics></math></div></div>
3277
3278 <h3 id='recognizing_weak_equivalences_quillens_theorem_a_and_b'>Recognizing weak equivalences: Quillen’s theorem A and B</h3>
3279
3280 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}, \mathcal{D}</annotation></semantics></math> be two <a class='existingWikiWord' href='/nlab/show/category'>categories</a> and let</p>
3281 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>
3282 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
3283
3284 </annotation></semantics></math></div>
3285 <p>be a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a> between them.</p>
3286
3287 <div class='num_theorem' id='QuillenTheoremA'>
3288 <h6 id='theorem'>Theorem</h6>
3289
3290 <p><strong>(<a href='#Quillen72'>Quillen 72</a>, theorem A)</strong></p>
3291
3292 <p>If for all <a class='existingWikiWord' href='/nlab/show/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>d \in \mathcal{D}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>/</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(F/d)\vert}</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/comma+category'>comma category</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>/</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>F/d</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible</a> (meaning that <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is a “homotopy <a class='existingWikiWord' href='/nlab/show/final+functor'>cofinal functor</a>”, hence a <a class='existingWikiWord' href='/nlab/show/final+%28infinity%2C1%29-functor'>cofinal (∞,1)-functor</a>), then</p>
3293 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mo>⟶</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒟</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>
3294 {\vert N(F) \vert}
3295 \;\colon\;
3296 {\vert N(\mathcal{C}) \vert}
3297 \longrightarrow
3298 {\vert N(\mathcal{D}) \vert}
3299
3300 </annotation></semantics></math></div>
3301 <p>is a <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a>.</p>
3302 </div>
3303
3304 <div class='num_theorem' id='QuillenTheoremB'>
3305 <h6 id='theorem_2'>Theorem</h6>
3306
3307 <p><strong>(<a href='#Quillen72'>Quillen 72</a> theorem B)</strong></p>
3308
3309 <p>If for all <a class='existingWikiWord' href='/nlab/show/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>d \in \mathcal{D}</annotation></semantics></math> we have that <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>/</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(F/d)\vert}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weakly homotopy equivalent</a> to a given <a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and all morphisms <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>f \colon d_1 \to d_2</annotation></semantics></math> induce <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalences</a> between these, then <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/fiber+sequence'>homotopy fiber</a> of <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(F) \vert}</annotation></semantics></math>, hence we have a <a class='existingWikiWord' href='/nlab/show/fiber+sequence'>homotopy fiber sequence</a> (in the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a>) of the form</p>
3310 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⟶</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mover><mo>⟶</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mover><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒟</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
3311 X
3312 \longrightarrow
3313 {\vert N(\mathcal{C}) \vert}
3314 \overset{\vert N(F) \vert }{\longrightarrow}
3315 {\vert N(\mathcal{D}) \vert}
3316 \,.
3317
3318 </annotation></semantics></math></div></div>
3319
3320 <p>As a consequence:</p>
3321
3322 <div class='num_prop'>
3323 <h6 id='proposition_2'>Proposition</h6>
3324
3325 <p><strong>(<a href='#McCord66'>McCord 66, theorem 6</a>, <a href='#Quillen78'>Quillen 78, prop. 1.6</a>)</strong></p>
3326
3327 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}, \mathcal{D}</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/partial+order'>posets</a> and consider <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>F \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a>.</p>
3328
3329 <p>If for each element/object <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>y \in \mathcal{D}</annotation></semantics></math> its <a class='existingWikiWord' href='/nlab/show/preimage'>preimage</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo>′</mo><mo>∈</mo><mi>Y</mi><mo stretchy='false'>|</mo><mi>y</mi><mo>′</mo><mo>≤</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^{-1}( \{ y&#39; \in Y \vert y&#39; \leq y \})</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible</a> geometric realization, then <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(F)\vert}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a>.</p>
3330 </div>
3331
3332 <p>An alternative proof is given in (<a href='#Barmak10'>Barmak 10</a>).</p>
3333
3334 <h3 id='natural_transformations_and_homotopies'>Natural transformations and homotopies</h3>
3335
3336 <div class='num_prop' id='NaturalTrafoMapsToHomotopy'>
3337 <h6 id='proposition_3'>Proposition</h6>
3338
3339 <p>A <a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>:</mo><mi>F</mi><mo>⇒</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>\eta : F \Rightarrow G</annotation></semantics></math> between two <a class='existingWikiWord' href='/nlab/show/functor'>functors</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>F, G : \mathcal{C} \to \mathcal{D}</annotation></semantics></math> induces under geometric realization a <a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a></p>
3340 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>η</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mo lspace='verythinmathspace'>:</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mo>⟶</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
3341 {|N(\eta)|} \colon {\vert N(F)\vert} \longrightarrow {\vert N(G) \vert}
3342 \,.
3343
3344 </annotation></semantics></math></div></div>
3345
3346 <div class='proof'>
3347 <h6 id='proof'>Proof</h6>
3348
3349 <p>The natural transformation is equivalently a functor of the form</p>
3350 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>𝒞</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy='false'>}</mo><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>
3351 \eta \;\colon\; \mathcal{C} \times \{0 \to 1\} \to \mathcal{D}
3352
3353 </annotation></semantics></math></div>
3354 <p>out of the <a class='existingWikiWord' href='/nlab/show/product+category'>product category</a> of <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/interval+category'>interval category</a>.</p>
3355
3356 <p>Since <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> of <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> preserves <a class='existingWikiWord' href='/nlab/show/cartesian+product'>Cartesian products</a> (see there) we have that</p>
3357 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thickmathspace'></mspace><msub><mo>≃</mo> <mi>iso</mi></msub><mspace width='thickmathspace'></mspace><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mo>×</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>
3358 {\vert N( \mathcal{C} \times \{0,1\} ) \vert}
3359 \;\simeq_{iso}\;
3360 {\vert N(\mathcal{C}) \vert} \times {\vert N(\{0 \to 1\}) \vert}
3361
3362 </annotation></semantics></math></div>
3363 <p>But this is a <a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a> in topological spaces, hence <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>η</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert N(\eta) \vert}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a>.</p>
3364 </div>
3365
3366 <div class='num_cor' id='RealizationOfEquivalenceIsHomotopyEquivalence'>
3367 <h6 id='corollary'>Corollary</h6>
3368
3369 <p>An <a class='existingWikiWord' href='/nlab/show/equivalence+of+categories'>equivalence of categories</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>≃</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C} \simeq \mathcal{D}</annotation></semantics></math> induces a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a> between their geometric realizations.</p>
3370 </div>
3371
3372 <div class='num_remark'>
3373 <h6 id='remark'>Remark</h6>
3374
3375 <p>The statement still remains true for a pair of <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functor</a>s <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>⇆</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C} \leftrightarrows \mathcal{D}</annotation></semantics></math>.</p>
3376 </div>
3377
3378 <div class='num_remark'>
3379 <h6 id='remark_2'>Remark</h6>
3380
3381 <p>Notice that the converse is far from true: Very different categories can have geometric realizations that are (weakly) homotopy equivalent. This is because geometric realization implicitly involves <a class='existingWikiWord' href='/nlab/show/Kan+fibrant+replacement'>Kan fibrant replacement</a>: it freely turns morphisms into <a class='existingWikiWord' href='/nlab/show/equivalence+in+an+%28infinity%2C1%29-category'>equivalences</a>.</p>
3382 </div>
3383
3384 <div class='num_cor' id='RealizationWithTerminalObjectIsContractible'>
3385 <h6 id='corollary_2'>Corollary</h6>
3386
3387 <p>If a <a class='existingWikiWord' href='/nlab/show/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/initial+object'>initial object</a> or a <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal object</a>, then its geometric realization is <a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible</a>.</p>
3388 </div>
3389
3390 <div class='proof'>
3391 <h6 id='proof_2'>Proof</h6>
3392
3393 <p>Assume the case of a terminal object, the other case works <a class='existingWikiWord' href='/nlab/show/duality'>formally dually</a>. Write <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>*</annotation></semantics></math> for the terminal category.</p>
3394
3395 <p>Then we have an equality of functors</p>
3396 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mo>*</mo></msub><mo>=</mo><mo stretchy='false'>(</mo><mo>*</mo><mover><mo>→</mo><mo>⊥</mo></mover><mi>C</mi><mo>→</mo><mo>*</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'>
3397 Id_* = (* \stackrel{\bottom}{\to} C \to *)
3398 \,,
3399
3400 </annotation></semantics></math></div>
3401 <p>where the first functor on the right picks the terminal object, and we have a <a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a></p>
3402 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub><mo>⇒</mo><mo stretchy='false'>(</mo><mi>C</mi><mo>→</mo><mo>*</mo><mover><mo>→</mo><mo>⊥</mo></mover><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
3403 Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C)
3404
3405 </annotation></semantics></math></div>
3406 <p>whose components are the unique morphisms into the terminal object.</p>
3407
3408 <p>By prop. <a class='maruku-ref' href='#NaturalTrafoMapsToHomotopy'>3</a> it follows that we have a <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mo>→</mo><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>\vert N(\mathcal{C}) \vert \to \vert N(\ast) \vert = \ast</annotation></semantics></math>.</p>
3409 </div>
3410
3411 <h3 id='behaviour_under_homotopy_colimits'>Behaviour under homotopy colimits</h3>
3412
3413 <div class='num_prop'>
3414 <h6 id='proposition_4'>Proposition</h6>
3415
3416 <p>For <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>𝒟</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>F \colon \mathcal{D} \to Cat</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a>, let</p>
3417 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>𝒟</mi><mover><mo>⟶</mo><mi>F</mi></mover><mi>Cat</mi><mover><mo>→</mo><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mover><mi>Top</mi></mrow><annotation encoding='application/x-tex'>
3418 {\vert N(F(-))\vert}
3419 \;\colon\;
3420 \mathcal{D}
3421 \overset{F}{\longrightarrow}
3422 Cat
3423 \stackrel{\vert N(-) \vert}{\to}
3424 Top
3425
3426 </annotation></semantics></math></div>
3427 <p>be its postcomposition with geometric realization of categories</p>
3428
3429 <p>Then we have a <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a></p>
3430 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>|</mo><mi>N</mi><mrow><mo>(</mo><mo>∫</mo><mi>F</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≃</mo><mi>hocolim</mi><mrow><mo stretchy='false'>|</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>N</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>
3431 {\left\vert N\left(\int F \right) \right\vert}
3432 \simeq
3433 hocolim {\vert F(N(-)) \vert}
3434
3435 </annotation></semantics></math></div>
3436 <p>exhibiting the <a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy colimit</a> in <a class='existingWikiWord' href='/nlab/show/Top'>Top</a> over <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>|</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo></mrow><annotation encoding='application/x-tex'>\vert N(F (-)) \vert</annotation></semantics></math> as the geometric realization of the <a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck construction</a> <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>\int F</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>.</p>
3437 </div>
3438
3439 <p>This is due to (<a href='#Thomason79'>Thomason 79</a>).</p>
3440
3441 <h2 id='related_concepts'>Related concepts</h2>
3442
3443 <ul>
3444 <li>
3445 <p><a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a></p>
3446
3447 <ul>
3448 <li><strong>of categories</strong>, <a class='existingWikiWord' href='/nlab/show/geometric+realization+of+simplicial+topological+spaces'>of simplicial topological spaces</a>, <a class='existingWikiWord' href='/nlab/show/geometric+realization+of+cohesive+infinity-groupoids'>of cohesive ∞-groupoids</a></li>
3449 </ul>
3450 </li>
3451 </ul>
3452
3453 <h2 id='references'>References</h2>
3454
3455 <h3 id='general'>General</h3>
3456
3457 <p>For general references see also <em><a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a></em> and <em><a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a></em>.</p>
3458
3459 <h3 id='quillens_theorems_a_and_b'>Quillen’s theorems A and B</h3>
3460
3461 <p>The original articles are</p>
3462
3463 <ul>
3464 <li id='McCord66'>
3465 <p><a class='existingWikiWord' href='/nlab/show/Michael+C.+McCord'>Michael C. McCord</a>, <em>Singular homology groups and homotopy groups of finite topological spaces</em>, Duke Math. J. 33 (1966), 465-474</p>
3466 </li>
3467
3468 <li id='Quillen72'>
3469 <p><a class='existingWikiWord' href='/nlab/show/Daniel+Quillen'>Daniel Quillen</a>, <em>Higher algebraic K-theory, I: Higher K-theories</em> Lect. Notes in Math. 341 (1972), 85-1 (<a href='http://math.mit.edu/~hrm/kansem/quillen-higher-algebraic-k-theory.pdf'>pdf</a>)</p>
3470 </li>
3471
3472 <li id='Quillen78'>
3473 <p><a class='existingWikiWord' href='/nlab/show/Daniel+Quillen'>Daniel Quillen</a>, <em>Homotopy properties of the poset of nontrivial p-subgroups of a group</em>, Adv. Math. 28 (1978), 101-128.</p>
3474 </li>
3475 </ul>
3476
3477 <p>The geometric realization of <a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck constructions</a> has been analyzed in</p>
3478
3479 <ul>
3480 <li id='Thomason79'><a class='existingWikiWord' href='/nlab/show/Robert+Thomason'>R. W. Thomason</a>, <em>Homotopy colimits in the category of small categories</em> , Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91109.</li>
3481 </ul>
3482
3483 <p>Review is in</p>
3484
3485 <ul>
3486 <li id='Barmak10'><a class='existingWikiWord' href='/nlab/show/Jonathan+Barmak'>Jonathan Barmak</a>, <em>On Quillen’s Theorem A for posets</em>, Journal of Combinatorial Theory Series A, Volume 118 Issue 8, November, 2011 Pages 2445-2453 (<a href='http://arxiv.org/abs/1005.0538'>arXiv:1005.0538</a>)</li>
3487 </ul>
3488
3489 <p>Further development includes</p>
3490
3491 <ul>
3492 <li>
3493 <p><a class='existingWikiWord' href='/nlab/show/Clark+Barwick'>Clark Barwick</a>, <a class='existingWikiWord' href='/nlab/show/Daniel+Kan'>Daniel Kan</a>, <em>A Quillen theorem <math class='maruku-mathml' display='inline' id='mathml_ea68c383f76618356121fcf995bd6e5db0f17093_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>B_n</annotation></semantics></math> for homotopy pullbacks</em> (<a href='http://arxiv.org/abs/1101.4879'>arXiv:1101.4879</a>)</p>
3494 </li>
3495
3496 <li>
3497 <p><a class='existingWikiWord' href='/nlab/show/David+Michael+Roberts'>David Roberts</a>, <em><a class='existingWikiWord' href='/davidroberts/show/Theorem+A+for+topological+categories' title='davidroberts'>Theorem A for topological categories</a></em></p>
3498 </li>
3499 </ul>
3500
3501 <p>
3502 </p>
3503
3504 <p>
3505
3506
3507
3508
3509
3510
3511
3512 </p> </div>
3513 </content>
3514 </entry>
3515 <entry>
3516 <title type="html">Borel model structure</title>
3517 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Borel+model+structure"/>
3518 <updated>2021-07-01T16:26:36Z</updated>
3519 <published>2014-04-15T05:41:15Z</published>
3520 <id>tag:ncatlab.org,2014-04-15:nLab,Borel+model+structure</id>
3521 <author>
3522 <name>Urs Schreiber</name>
3523 </author>
3524 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Borel+model+structure">
3525 <div xmlns="http://www.w3.org/1999/xhtml">
3526 <div class='rightHandSide'>
3527 <div class='toc clickDown' tabindex='0'>
3528 <h3 id='context'>Context</h3>
3529
3530 <h4 id='model_category_theory'>Model category theory</h4>
3531
3532 <div class='hide'>
3533 <p><strong><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></strong></p>
3534
3535 <h2 id='definitions'>Definitions</h2>
3536
3537 <ul>
3538 <li>
3539 <p><a class='existingWikiWord' href='/nlab/show/category+with+weak+equivalences'>category with weak equivalences</a></p>
3540 </li>
3541
3542 <li>
3543 <p><a class='existingWikiWord' href='/nlab/show/weak+factorization+system'>weak factorization system</a></p>
3544 </li>
3545
3546 <li>
3547 <p><a class='existingWikiWord' href='/nlab/show/homotopy+%28as+an+operation%29'>homotopy</a></p>
3548
3549 <ul>
3550 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></li>
3551 </ul>
3552 </li>
3553
3554 <li>
3555 <p><a class='existingWikiWord' href='/nlab/show/small+object+argument'>small object argument</a></p>
3556 </li>
3557
3558 <li>
3559 <p><a class='existingWikiWord' href='/nlab/show/resolution'>resolution</a></p>
3560 </li>
3561 </ul>
3562
3563 <h2 id='morphisms'>Morphisms</h2>
3564
3565 <ul>
3566 <li>
3567 <p><a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>Quillen adjunction</a></p>
3568
3569 <ul>
3570 <li>
3571 <p><a class='existingWikiWord' href='/nlab/show/Quillen+equivalence'>Quillen equivalence</a></p>
3572 </li>
3573
3574 <li>
3575 <p><a class='existingWikiWord' href='/nlab/show/Quillen+bifunctor'>Quillen bifunctor</a></p>
3576 </li>
3577
3578 <li>
3579 <p><a class='existingWikiWord' href='/nlab/show/derived+functor'>derived functor</a></p>
3580 </li>
3581 </ul>
3582 </li>
3583 </ul>
3584
3585 <h2 id='universal_constructions'>Universal constructions</h2>
3586
3587 <ul>
3588 <li>
3589 <p><a class='existingWikiWord' href='/nlab/show/homotopy+Kan+extension'>homotopy Kan extension</a></p>
3590 </li>
3591
3592 <li>
3593 <p><a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy limit</a>/<a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy colimit</a></p>
3594 </li>
3595
3596 <li>
3597 <p><a class='existingWikiWord' href='/nlab/show/Bousfield-Kan+map'>Bousfield-Kan map</a></p>
3598 </li>
3599 </ul>
3600
3601 <h2 id='refinements'>Refinements</h2>
3602
3603 <ul>
3604 <li>
3605 <p><a class='existingWikiWord' href='/nlab/show/monoidal+model+category'>monoidal model category</a></p>
3606
3607 <ul>
3608 <li><a class='existingWikiWord' href='/nlab/show/monoidal+Quillen+adjunction'>monoidal Quillen adjunction</a></li>
3609 </ul>
3610 </li>
3611
3612 <li>
3613 <p><a class='existingWikiWord' href='/nlab/show/enriched+model+category'>enriched model category</a></p>
3614
3615 <ul>
3616 <li><a class='existingWikiWord' href='/nlab/show/enriched+Quillen+adjunction'>enriched Quillen adjunction</a></li>
3617 </ul>
3618 </li>
3619
3620 <li>
3621 <p><a class='existingWikiWord' href='/nlab/show/simplicial+model+category'>simplicial model category</a></p>
3622
3623 <ul>
3624 <li><a class='existingWikiWord' href='/nlab/show/simplicial+Quillen+adjunction'>simplicial Quillen adjunction</a></li>
3625 </ul>
3626 </li>
3627
3628 <li>
3629 <p><a class='existingWikiWord' href='/nlab/show/cofibrantly+generated+model+category'>cofibrantly generated model category</a></p>
3630
3631 <ul>
3632 <li>
3633 <p><a class='existingWikiWord' href='/nlab/show/combinatorial+model+category'>combinatorial model category</a></p>
3634 </li>
3635
3636 <li>
3637 <p><a class='existingWikiWord' href='/nlab/show/cellular+model+category'>cellular model category</a></p>
3638 </li>
3639 </ul>
3640 </li>
3641
3642 <li>
3643 <p><a class='existingWikiWord' href='/nlab/show/algebraic+model+category'>algebraic model category</a></p>
3644 </li>
3645
3646 <li>
3647 <p><a class='existingWikiWord' href='/nlab/show/compactly+generated+model+category'>compactly generated model category</a></p>
3648 </li>
3649
3650 <li>
3651 <p><a class='existingWikiWord' href='/nlab/show/proper+model+category'>proper model category</a></p>
3652 </li>
3653
3654 <li>
3655 <p><a class='existingWikiWord' href='/nlab/show/cartesian+model+category'>cartesian closed model category</a>, <a class='existingWikiWord' href='/nlab/show/locally+cartesian+closed+model+category'>locally cartesian closed model category</a></p>
3656 </li>
3657
3658 <li>
3659 <p><a class='existingWikiWord' href='/nlab/show/stable+model+category'>stable model category</a></p>
3660 </li>
3661 </ul>
3662
3663 <h2 id='producing_new_model_structures'>Producing new model structures</h2>
3664
3665 <ul>
3666 <li>
3667 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+functors'>on functor categories (global)</a></p>
3668
3669 <ul>
3670 <li><a class='existingWikiWord' href='/nlab/show/Reedy+model+structure'>Reedy model structure</a></li>
3671 </ul>
3672 </li>
3673
3674 <li>
3675 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+an+over+category'>on overcategories</a></p>
3676 </li>
3677
3678 <li>
3679 <p><a class='existingWikiWord' href='/nlab/show/Bousfield+localization+of+model+categories'>Bousfield localization</a></p>
3680 </li>
3681
3682 <li>
3683 <p><a class='existingWikiWord' href='/nlab/show/transferred+model+structure'>transferred model structure</a></p>
3684
3685 <ul>
3686 <li><a class='existingWikiWord' href='/nlab/show/model+structure+on+algebraic+fibrant+objects'>model structure on algebraic fibrant objects</a></li>
3687 </ul>
3688 </li>
3689
3690 <li>
3691 <p><a class='existingWikiWord' href='/nlab/show/Grothendieck+construction+for+model+categories'>Grothendieck construction for model categories</a></p>
3692 </li>
3693 </ul>
3694
3695 <h2 id='presentation_of_categories'>Presentation of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories</h2>
3696
3697 <ul>
3698 <li>
3699 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
3700 </li>
3701
3702 <li>
3703 <p><a class='existingWikiWord' href='/nlab/show/simplicial+localization'>simplicial localization</a></p>
3704 </li>
3705
3706 <li>
3707 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categorical+hom-space'>(∞,1)-categorical hom-space</a></p>
3708 </li>
3709
3710 <li>
3711 <p><a class='existingWikiWord' href='/nlab/show/locally+presentable+%28infinity%2C1%29-category'>presentable (∞,1)-category</a></p>
3712 </li>
3713 </ul>
3714
3715 <h2 id='model_structures'>Model structures</h2>
3716
3717 <ul>
3718 <li><a class='existingWikiWord' href='/nlab/show/Cisinski+model+structure'>Cisinski model structure</a></li>
3719 </ul>
3720
3721 <h3 id='for_groupoids'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</h3>
3722
3723 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+infinity-groupoids'>for ∞-groupoids</a></p>
3724
3725 <ul>
3726 <li>
3727 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+topological+spaces'>on topological spaces</a></p>
3728
3729 <ul>
3730 <li>
3731 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical model structure</a></p>
3732 </li>
3733
3734 <li>
3735 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+Delta-generated+topological+spaces'>on Delta-generated spaces</a></p>
3736 </li>
3737
3738 <li>
3739 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+diffeological+spaces'>on diffeological spaces</a></p>
3740 </li>
3741
3742 <li>
3743 <p><a class='existingWikiWord' href='/nlab/show/Str%C3%B8m+model+structure'>Strom model structure</a></p>
3744 </li>
3745 </ul>
3746 </li>
3747
3748 <li>
3749 <p><a class='existingWikiWord' href='/nlab/show/Thomason+model+structure'>Thomason model structure</a></p>
3750 </li>
3751
3752 <li>
3753 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presheaves+over+a+test+category'>model structure on presheaves over a test category</a></p>
3754 </li>
3755
3756 <li>
3757 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+sets'>on simplicial sets</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+on+semi-simplicial+sets'>on semi-simplicial sets</a></p>
3758
3759 <ul>
3760 <li>
3761 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+simplicial+sets'>classical model structure</a></p>
3762 </li>
3763
3764 <li>
3765 <p><a class='existingWikiWord' href='/nlab/show/constructive+model+structure+on+simplicial+sets'>constructive model structure</a></p>
3766 </li>
3767
3768 <li>
3769 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+left+fibrations'>for right/left fibrations</a></p>
3770 </li>
3771 </ul>
3772 </li>
3773
3774 <li>
3775 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+groupoids'>model structure on simplicial groupoids</a></p>
3776 </li>
3777
3778 <li>
3779 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cubical+sets'>on cubical sets</a></p>
3780 </li>
3781
3782 <li>
3783 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+strict+omega-groupoids'>on strict ∞-groupoids</a>, <a class='existingWikiWord' href='/nlab/show/canonical+model+structure+on+groupoids'>on groupoids</a></p>
3784 </li>
3785
3786 <li>
3787 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+chain+complexes'>on chain complexes</a>/<a class='existingWikiWord' href='/nlab/show/model+structure+on+cosimplicial+abelian+groups'>model structure on cosimplicial abelian groups</a></p>
3788
3789 <p>related by the <a class='existingWikiWord' href='/nlab/show/Dold-Kan+correspondence'>Dold-Kan correspondence</a></p>
3790 </li>
3791
3792 <li>
3793 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cosimplicial+simplicial+sets'>model structure on cosimplicial simplicial sets</a></p>
3794 </li>
3795 </ul>
3796
3797 <h3 id='for_rational_groupoids'>for rational <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</h3>
3798
3799 <ul>
3800 <li>
3801 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-algebras'>model structure on dgc-algebras</a></p>
3802 </li>
3803
3804 <li>
3805 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+equivariant+dgc-algebras'>model structure on equivariant dgc-algebras</a></p>
3806
3807 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+equivariant+chain+complexes'>model structure on equivariant chain complexes</a></p>
3808 </li>
3809 </ul>
3810
3811 <h3 id='for_groupoids_2'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-groupoids</h3>
3812
3813 <ul>
3814 <li>
3815 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+homotopy+n-types'>for n-groupoids</a>/<a class='existingWikiWord' href='/nlab/show/model+structure+for+homotopy+n-types'>for n-types</a></p>
3816 </li>
3817
3818 <li>
3819 <p><a class='existingWikiWord' href='/nlab/show/canonical+model+structure+on+groupoids'>for 1-groupoids</a></p>
3820 </li>
3821 </ul>
3822
3823 <h3 id='for_groups'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groups</h3>
3824
3825 <ul>
3826 <li>
3827 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+groups'>model structure on simplicial groups</a></p>
3828 </li>
3829
3830 <li>
3831 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+reduced+simplicial+sets'>model structure on reduced simplicial sets</a></p>
3832 </li>
3833 </ul>
3834
3835 <h3 id='for_algebras'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebras</h3>
3836
3837 <h4 id='general'>general</h4>
3838
3839 <ul>
3840 <li>
3841 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+monoids+in+a+monoidal+model+category'>on monoids</a></p>
3842 </li>
3843
3844 <li>
3845 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+algebras'>on simplicial T-algebras</a>, on <a class='existingWikiWord' href='/nlab/show/homotopy+T-algebra'>homotopy T-algebra</a>s</p>
3846 </li>
3847
3848 <li>
3849 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algebras+over+a+monad'>on algebas over a monad</a></p>
3850 </li>
3851
3852 <li>
3853 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad'>on modules over an algebra over an operad</a></p>
3854 </li>
3855 </ul>
3856
3857 <h4 id='specific'>specific</h4>
3858
3859 <ul>
3860 <li>
3861 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-algebras'>model structure on differential-graded commutative algebras</a></p>
3862 </li>
3863
3864 <li>
3865 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+differential+graded-commutative+superalgebras'>model structure on differential graded-commutative superalgebras</a></p>
3866 </li>
3867
3868 <li>
3869 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-algebras+over+an+operad'>on dg-algebras over an operad</a></p>
3870
3871 <ul>
3872 <li>
3873 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-algebras'>on dg-algebras</a> and on <a class='existingWikiWord' href='/nlab/show/simplicial+ring'>on simplicial rings</a>/<a class='existingWikiWord' href='/nlab/show/model+structure+on+cosimplicial+rings'>on cosimplicial rings</a></p>
3874
3875 <p>related by the <a class='existingWikiWord' href='/nlab/show/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p>
3876 </li>
3877
3878 <li>
3879 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+L-infinity+algebras'>for L-∞ algebras</a>: <a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-Lie+algebras'>on dg-Lie algebras</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-coalgebras'>on dg-coalgebras</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+Lie+algebras'>on simplicial Lie algebras</a></p>
3880 </li>
3881 </ul>
3882 </li>
3883
3884 <li>
3885 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-modules'>model structure on dg-modules</a></p>
3886 </li>
3887 </ul>
3888
3889 <h3 id='for_stablespectrum_objects'>for stable/spectrum objects</h3>
3890
3891 <ul>
3892 <li>
3893 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+spectra'>model structure on spectra</a></p>
3894 </li>
3895
3896 <li>
3897 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+ring+spectra'>model structure on ring spectra</a></p>
3898 </li>
3899
3900 <li>
3901 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presheaves+of+spectra'>model structure on presheaves of spectra</a></p>
3902 </li>
3903 </ul>
3904
3905 <h3 id='for_categories'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories</h3>
3906
3907 <ul>
3908 <li>
3909 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+relative+categories'>on categories with weak equivalences</a></p>
3910 </li>
3911
3912 <li>
3913 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+quasi-categories'>Joyal model for quasi-categories</a></p>
3914 </li>
3915
3916 <li>
3917 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+sSet-categories'>on sSet-categories</a></p>
3918 </li>
3919
3920 <li>
3921 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+complete+Segal+spaces'>for complete Segal spaces</a></p>
3922 </li>
3923
3924 <li>
3925 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+Cartesian+fibrations'>for Cartesian fibrations</a></p>
3926 </li>
3927 </ul>
3928
3929 <h3 id='for_stable_categories'>for stable <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories</h3>
3930
3931 <ul>
3932 <li><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-categories'>on dg-categories</a></li>
3933 </ul>
3934
3935 <h3 id='for_operads'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-operads</h3>
3936
3937 <ul>
3938 <li>
3939 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+operads'>on operads</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+for+Segal+operads'>for Segal operads</a></p>
3940
3941 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad'>on modules over an algebra over an operad</a></p>
3942 </li>
3943
3944 <li>
3945 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dendroidal+sets'>on dendroidal sets</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces'>for dendroidal complete Segal spaces</a>, <a class='existingWikiWord' href='/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations'>for dendroidal Cartesian fibrations</a></p>
3946 </li>
3947 </ul>
3948
3949 <h3 id='for_categories_2'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n,r)</annotation></semantics></math>-categories</h3>
3950
3951 <ul>
3952 <li>
3953 <p><a class='existingWikiWord' href='/nlab/show/Theta-space'>for (n,r)-categories as ∞-spaces</a></p>
3954 </li>
3955
3956 <li>
3957 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+weak+complicial+sets'>for weak ∞-categories as weak complicial sets</a></p>
3958 </li>
3959
3960 <li>
3961 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cellular+sets'>on cellular sets</a></p>
3962 </li>
3963
3964 <li>
3965 <p><a class='existingWikiWord' href='/nlab/show/canonical+model+structure'>on higher categories in general</a></p>
3966 </li>
3967
3968 <li>
3969 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+strict+omega-categories'>on strict ∞-categories</a></p>
3970 </li>
3971 </ul>
3972
3973 <h3 id='for_sheaves__stacks'>for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-sheaves / <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks</h3>
3974
3975 <ul>
3976 <li>
3977 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+homotopical+presheaves'>on homotopical presheaves</a></p>
3978
3979 <ul>
3980 <li>
3981 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>on simplicial presheaves</a></p>
3982
3983 <p><a class='existingWikiWord' href='/nlab/show/global+model+structure+on+simplicial+presheaves'>global model structure</a>/<a class='existingWikiWord' href='/nlab/show/%C4%8Cech+model+structure+on+simplicial+presheaves'>Cech model structure</a>/<a class='existingWikiWord' href='/nlab/show/local+model+structure+on+simplicial+presheaves'>local model structure</a></p>
3984
3985 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+sheaves'>on simplicial sheaves</a></p>
3986
3987 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids'>on presheaves of simplicial groupoids</a></p>
3988
3989 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+sSet-enriched+presheaves'>on sSet-enriched presheaves</a></p>
3990 </li>
3991 </ul>
3992 </li>
3993
3994 <li>
3995 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+%282%2C1%29-sheaves'>model structure for (2,1)-sheaves</a>/for stacks</p>
3996 </li>
3997 </ul>
3998 <div>
3999 <p>
4000 <a href='/nlab/edit/model+category+theory+-+contents'>Edit this sidebar</a>
4001 </p>
4002 </div></div>
4003
4004 <h4 id='group_theory'>Group Theory</h4>
4005
4006 <div class='hide'>
4007 <p><strong><a class='existingWikiWord' href='/nlab/show/group+theory'>group theory</a></strong></p>
4008
4009 <ul>
4010 <li><a class='existingWikiWord' href='/nlab/show/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-group</a></li>
4011
4012 <li><a class='existingWikiWord' href='/nlab/show/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li>
4013
4014 <li><a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></li>
4015
4016 <li><a class='existingWikiWord' href='/nlab/show/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/infinity-action'>∞-action</a></li>
4017
4018 <li><a class='existingWikiWord' href='/nlab/show/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/infinity-representation'>∞-representation</a></li>
4019
4020 <li><a class='existingWikiWord' href='/nlab/show/progroup'>progroup</a></li>
4021
4022 <li><a class='existingWikiWord' href='/nlab/show/homogeneous+space'>homogeneous space</a></li>
4023 </ul>
4024
4025 <h3 id='classical_groups'>Classical groups</h3>
4026
4027 <ul>
4028 <li>
4029 <p><a class='existingWikiWord' href='/nlab/show/general+linear+group'>general linear group</a></p>
4030 </li>
4031
4032 <li>
4033 <p><a class='existingWikiWord' href='/nlab/show/unitary+group'>unitary group</a></p>
4034
4035 <ul>
4036 <li><a class='existingWikiWord' href='/nlab/show/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/projective+unitary+group'>projective unitary group</a></li>
4037 </ul>
4038 </li>
4039
4040 <li>
4041 <p><a class='existingWikiWord' href='/nlab/show/orthogonal+group'>orthogonal group</a></p>
4042
4043 <ul>
4044 <li><a class='existingWikiWord' href='/nlab/show/special+orthogonal+group'>special orthogonal group</a></li>
4045 </ul>
4046 </li>
4047
4048 <li>
4049 <p><a class='existingWikiWord' href='/nlab/show/symplectic+group'>symplectic group</a></p>
4050 </li>
4051 </ul>
4052
4053 <h3 id='finite_groups'>Finite groups</h3>
4054
4055 <ul>
4056 <li>
4057 <p><a class='existingWikiWord' href='/nlab/show/finite+group'>finite group</a></p>
4058 </li>
4059
4060 <li>
4061 <p><a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/braid+group'>braid group</a></p>
4062 </li>
4063
4064 <li>
4065 <p><a class='existingWikiWord' href='/nlab/show/classification+of+finite+simple+groups'>classification of finite simple groups</a></p>
4066 </li>
4067
4068 <li>
4069 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+group'>sporadic finite simple groups</a></p>
4070
4071 <ul>
4072 <li><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu group</a></li>
4073 </ul>
4074 </li>
4075 </ul>
4076
4077 <h3 id='group_schemes'>Group schemes</h3>
4078
4079 <ul>
4080 <li><a class='existingWikiWord' href='/nlab/show/algebraic+group'>algebraic group</a></li>
4081
4082 <li><a class='existingWikiWord' href='/nlab/show/abelian+variety'>abelian variety</a></li>
4083 </ul>
4084
4085 <h3 id='topological_groups'>Topological groups</h3>
4086
4087 <ul>
4088 <li>
4089 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topological group</a></p>
4090 </li>
4091
4092 <li>
4093 <p><a class='existingWikiWord' href='/nlab/show/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/locally+compact+topological+group'>locally compact topological group</a></p>
4094 </li>
4095
4096 <li>
4097 <p><a class='existingWikiWord' href='/nlab/show/maximal+compact+subgroup'>maximal compact subgroup</a></p>
4098 </li>
4099
4100 <li>
4101 <p><a class='existingWikiWord' href='/nlab/show/string+group'>string group</a></p>
4102 </li>
4103 </ul>
4104
4105 <h3 id='lie_groups'>Lie groups</h3>
4106
4107 <ul>
4108 <li>
4109 <p><a class='existingWikiWord' href='/nlab/show/Lie+group'>Lie group</a></p>
4110 </li>
4111
4112 <li>
4113 <p><a class='existingWikiWord' href='/nlab/show/compact+Lie+group'>compact Lie group</a></p>
4114 </li>
4115
4116 <li>
4117 <p><a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moody group</a></p>
4118 </li>
4119 </ul>
4120
4121 <h3 id='superlie_groups'>Super-Lie groups</h3>
4122
4123 <ul>
4124 <li>
4125 <p><a class='existingWikiWord' href='/nlab/show/supergroup'>super Lie group</a></p>
4126 </li>
4127
4128 <li>
4129 <p><a class='existingWikiWord' href='/nlab/show/super+Euclidean+group'>super Euclidean group</a></p>
4130 </li>
4131 </ul>
4132
4133 <h3 id='higher_groups'>Higher groups</h3>
4134
4135 <ul>
4136 <li>
4137 <p><a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a></p>
4138
4139 <ul>
4140 <li><a class='existingWikiWord' href='/nlab/show/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/strict+2-group'>strict 2-group</a></li>
4141 </ul>
4142 </li>
4143
4144 <li>
4145 <p><a class='existingWikiWord' href='/nlab/show/n-group'>n-group</a></p>
4146 </li>
4147
4148 <li>
4149 <p><a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-group</a></p>
4150
4151 <ul>
4152 <li>
4153 <p><a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a></p>
4154 </li>
4155
4156 <li>
4157 <p><a class='existingWikiWord' href='/nlab/show/crossed+complex'>crossed complex</a></p>
4158 </li>
4159
4160 <li>
4161 <p><a class='existingWikiWord' href='/nlab/show/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p>
4162 </li>
4163
4164 <li>
4165 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></p>
4166 </li>
4167 </ul>
4168 </li>
4169
4170 <li>
4171 <p><a class='existingWikiWord' href='/nlab/show/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/fivebrane+6-group'>fivebrane Lie 6-group</a></p>
4172 </li>
4173 </ul>
4174
4175 <h3 id='cohomology_and_extensions'>Cohomology and Extensions</h3>
4176
4177 <ul>
4178 <li>
4179 <p><a class='existingWikiWord' href='/nlab/show/group+cohomology'>group cohomology</a></p>
4180 </li>
4181
4182 <li>
4183 <p><a class='existingWikiWord' href='/nlab/show/group+extension'>group extension</a>,</p>
4184 </li>
4185
4186 <li>
4187 <p><a class='existingWikiWord' href='/nlab/show/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/Ext'>Ext-group</a></p>
4188 </li>
4189 </ul>
4190
4191 <h3 id='_related_concepts'>Related concepts</h3>
4192
4193 <ul>
4194 <li><a class='existingWikiWord' href='/nlab/show/quantum+group'>quantum group</a></li>
4195 </ul>
4196 <div>
4197 <p>
4198 <a href='/nlab/edit/group+theory+-+contents'>Edit this sidebar</a>
4199 </p>
4200 </div></div>
4201 </div>
4202 </div>
4203
4204 <h1 id='contents'>Contents</h1>
4205 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#CofibrantReplacementAndHomotopyQuotientsFixedPoints'>Cofibrant replacement and homotopy quotients/fixed points</a></li><li><a href='#RelationToSliceOverSimplicialClassifyingSpace'>Relation to the slice over the simplicial classifying space</a></li><li><a href='#RelationToModelStructureOnPlainSimplicialSets'>Relation to the model structure on plain simplicial sets</a></li><li><a href='#relation_to_the_fine_model_structure_of_equivariant_homotopy_theory'>Relation to the fine model structure of equivariant homotopy theory</a></li><li><a href='#GeneralizationToSimplicialPresheaves'>Generalization to simplicial presheaves</a></li></ul></li><li><a href='#references'>References</a></li></ul></div>
4206 <h2 id='idea'>Idea</h2>
4207
4208 <p>Given a <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math>, the <em>Borel model structure</em> is a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> structure on the <a class='existingWikiWord' href='/nlab/show/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> equipped with <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/action'>action</a> which presents the <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a> of <a class='existingWikiWord' href='/nlab/show/infinity-action'>∞-actions</a> of the <a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-group</a> (see there) presented by <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p>
4209
4210 <p>In the context of <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant homotopy theory</a> this is also called the “coarse model structure” (e.g. <a href='#Guillou'>Guillou, section 5</a>), since it is not equivalent to the “fine” homotopy theory of <a class='existingWikiWord' href='/nlab/show/topological+G-space'>G-spaces</a> which enters <a class='existingWikiWord' href='/nlab/show/Elmendorf%27s+theorem'>Elmendorf&#39;s theorem</a>.</p>
4211
4212 <h2 id='definition'>Definition</h2>
4213
4214 <p>\begin{defn}\label{BorelModelStructure}</p>
4215
4216 <p>For <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a> write</p>
4217
4218 <ul>
4219 <li>
4220 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathbf{B}G_\bullet</annotation></semantics></math> for the one-object <a class='existingWikiWord' href='/nlab/show/simplicially+enriched+category'>sSet-enriched category</a> (here: a <a class='existingWikiWord' href='/nlab/show/simplicial+groupoid'>simplicial groupoid</a>) whose <a class='existingWikiWord' href='/nlab/show/hom-object'>hom-object</a> is <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math>.</p>
4221 </li>
4222
4223 <li>
4224 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>Actions</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>G_\bullet Actions(sSet) \;\coloneqq\; sSetCat\big(\mathbf{B}G_\bullet, sSet\big)</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/enriched+functor+category'>enriched functor category</a> to <a class='existingWikiWord' href='/nlab/show/SimpSet'>SimplicialSets</a>.</p>
4225 </li>
4226
4227 <li>
4228 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sSet</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub><mo>≔</mo><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>G_\bullet Acts(sSet)_{proj} \coloneqq sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}</annotation></semantics></math> for the projective <a class='existingWikiWord' href='/nlab/show/model+structure+on+functors'>model structure on functors</a> (projective <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a>).</p>
4229 </li>
4230 </ul>
4231
4232 <p>This is the <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math> <em>Borel model structure</em>, naturally a <a class='existingWikiWord' href='/nlab/show/simplicial+model+category'>simplicial model category</a> (<a href='#DDK80'>DDK 80, Prop. 2.4</a>, <a href='#GoerssJardine09'>Goerss &amp; Jardine 09, Chapter V, Thm. 2.3</a>).</p>
4233
4234 <p>\end{defn}</p>
4235
4236 <h2 id='properties'>Properties</h2>
4237
4238 <h3 id='CofibrantReplacementAndHomotopyQuotientsFixedPoints'>Cofibrant replacement and homotopy quotients/fixed points</h3>
4239
4240 <p>\begin{prop}\label{CofibrationsOfSimplicialActions} <strong>(cofibrations of simplicial actions)</strong> \linebreak The cofibrations <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>i \colon X \to Y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}</annotation></semantics></math> (Def. \ref{BorelModelStructure}) are precisely those morphisms such that</p>
4241
4242 <ol>
4243 <li>
4244 <p>the underlying morphism of <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> is a <a class='existingWikiWord' href='/nlab/show/monomorphism'>monomorphism</a>;</p>
4245 </li>
4246
4247 <li>
4248 <p>the <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/action'>action</a> is a relatively <a class='existingWikiWord' href='/nlab/show/free+action'>free action</a>, i.e. <a class='existingWikiWord' href='/nlab/show/free+action'>free</a> on all <a class='existingWikiWord' href='/nlab/show/simplex'>simplices</a> not in the <a class='existingWikiWord' href='/nlab/show/image'>image</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>.</p>
4249 </li>
4250 </ol>
4251
4252 <p>\end{prop}</p>
4253
4254 <p>This is (<a href='#DDK80'>DDK 80, Prop. 2.2. (ii)</a>, <a href='#Guillou'>Guillou, Prop. 5.3</a>, <a href='#GoerssJardine09'>Goerss &amp; Jardine 09, V Lem. 2.4</a>).</p>
4255
4256 <p>\begin{remark} In particular this means that an object is <a class='existingWikiWord' href='/nlab/show/fibrant+object'>cofibrant</a> in <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}</annotation></semantics></math> if the <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/action'>action</a> on it is <a class='existingWikiWord' href='/nlab/show/free+action'>free</a>.</p>
4257
4258 <p>Hence <a class='existingWikiWord' href='/nlab/show/fibrant+replacement'>cofibrant replacement</a> is obtained by forming the <a class='existingWikiWord' href='/nlab/show/cartesian+product'>product</a> with the model <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>W G_\bullet</annotation></semantics></math> for the total space of the <a class='existingWikiWord' href='/nlab/show/universal+principal+bundle'>universal principal bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math> (see at <em><a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a></em> for notation and more details). \end{remark}</p>
4259
4260 <p>\begin{remark} It follows that for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>X, A \in sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categorical+hom-space'>derived hom space</a></p>
4261 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><msub><mi>Hom</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
4262 R Hom_G(X,A)
4263
4264 </annotation></semantics></math></div>
4265 <p>models the Borel <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/equivariant+cohomology'>equivariant cohomology</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/coefficient'>coefficients</a> in <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p>
4266
4267 <p>In particular,if <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/fibrant+object'>fibrant</a> (the underlying simplicial set is a <a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complex</a>) then</p>
4268
4269 <ol>
4270 <li>
4271 <p>if the <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>G_\bullet</annotation></semantics></math>-action on <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is trivial, then</p>
4272 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><msub><mi>Hom</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>W</mi><mi>G</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy='false'>(</mo><mi>W</mi><mi>G</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
4273 R Hom_G(X,A)
4274 \simeq
4275 Hom_G(W G \times X , A)
4276 \simeq
4277 Hom(W G \times_G X, A)
4278
4279 </annotation></semantics></math></div>
4280 <p>is equivalently maps of <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> out of the <a class='existingWikiWord' href='/nlab/show/Borel+construction'>Borel construction</a> on <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>;</p>
4281 </li>
4282
4283 <li>
4284 <p>if <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>X = \ast </annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/point'>point</a> then</p>
4285 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><msub><mi>Hom</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>W</mi><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy='false'>(</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>A</mi> <mrow><mi>h</mi><mi>G</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>
4286 R Hom_G(X,A)
4287 \simeq
4288 Hom_G(W G, A)
4289 \simeq
4290 Hom(\overline{W} G , A)
4291 \simeq
4292 A^{h G}
4293
4294 </annotation></semantics></math></div>
4295 <p>is the <a class='existingWikiWord' href='/nlab/show/homotopy+fixed+point'>homotopy fixed points</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p>
4296 </li>
4297 </ol>
4298
4299 <p>\end{remark}</p>
4300
4301 <h3 id='RelationToSliceOverSimplicialClassifyingSpace'>Relation to the slice over the simplicial classifying space</h3>
4302
4303 <p>\begin{prop}\label{QuillenEquivalenceToSliceOverSimplicialClassifyingSpace} For <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a>, there is a pair of <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functors</a></p>
4304 <div class='maruku-equation' id='eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sSet</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><mi>G</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mi>G</mi></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub><mi>W</mi><mi>G</mi></mrow></mover></munderover><msub><mi>sSet</mi> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>
4305
4306 G_\bullet Acts(sSet)_{proj}
4307 \underoverset
4308 {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}}
4309 {\overset{ (-) \times_{\overline{W}G} W G }{\longleftarrow}}
4310 {\bot}
4311 sSet_{/\overline{W}G}
4312
4313 </annotation></semantics></math></div>
4314 <p>which constitute a <a class='existingWikiWord' href='/nlab/show/simplicial+Quillen+adjunction'>simplicial</a> <a class='existingWikiWord' href='/nlab/show/Quillen+equivalence'>Quillen equivalence</a> between the Borel model structure (Def. \ref{BorelModelStructure}) and the <a class='existingWikiWord' href='/nlab/show/model+structure+on+an+over+category'>slice model structure</a> of the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a> slices over the <a class='existingWikiWord' href='/nlab/show/simplicial+classifying+space'>simplicial classifying space</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding='application/x-tex'>\overline{W}G</annotation></semantics></math>.,</p>
4315
4316 <p>\end{prop}</p>
4317
4318 <p>(<a href='#DDK80'>DDK 80, Prop. 2.3, Prop. 2.4</a>) Here:</p>
4319
4320 <ul>
4321 <li>
4322 <p>the <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right adjoint</a> forms <a class='existingWikiWord' href='/nlab/show/associated+bundle'>associated bundles</a> to <a class='existingWikiWord' href='/nlab/show/universal+principal+bundle'>universal principal bundles</a></p>
4323 </li>
4324
4325 <li>
4326 <p>the <a class='existingWikiWord' href='/nlab/show/left+adjoint'>left adjoint</a> forms <a class='existingWikiWord' href='/nlab/show/fiber+sequence'>homotopy fibers</a>.</p>
4327 </li>
4328 </ul>
4329
4330 <p>In fact, these are <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/enriched+functor'>enriched functors</a> which induced an <a class='existingWikiWord' href='/nlab/show/equivalence+of+%28infinity%2C1%29-categories'>equivalence of (infinity,1)-categories</a> between the <a class='existingWikiWord' href='/nlab/show/simplicial+localization'>simplicial localizations</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mi>sSetCat</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><mo>,</mo><mi>sSet</mi><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi></msub><mo>≃</mo><msub><mi>L</mi> <mi>W</mi></msub><msub><mi>sSet</mi> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>H</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>L_W sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} \simeq L_W sSet_{/\overline{W}H}</annotation></semantics></math> (<a href='#DDK80'>DDK 80, Prop. 2.5</a>).</p>
4331
4332 <p>This kind of relation is discussed in more detail at <em><a class='existingWikiWord' href='/nlab/show/infinity-action'>∞-action</a></em>.</p>
4333
4334 <p>\begin{remark}\label{sSetEnrichmentOfAdjunctionToSliceOverSimpClassSpace} <strong>(sSet-enrichement of the adjunction)</strong> \linebreak The statement that <a class='maruku-eqref' href='#eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace'>(1)</a> is an <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>-<em><a class='existingWikiWord' href='/nlab/show/enriched+adjunction'>enriched adjunction</a></em> is not made explicit in <a href='#DDK80'>DDK 80</a>; there it only says that the functors form a plain <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjunction</a> (<a href='#DDK80'>DDK 80, Prop. 2.3</a>) and that they are each <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/enriched+functor'>enriched functors</a> (<a href='#DDK80'>DDK 80, Prop. 2.4</a>).</p>
4335
4336 <p>The remaining observation that we have a <a class='existingWikiWord' href='/nlab/show/natural+isomorphism'>natural isomorphism</a> of <a class='existingWikiWord' href='/nlab/show/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/hom-object'>hom-objects</a></p>
4337 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>[</mo><mi>X</mi><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub><mi>W</mi><mi>G</mi><mo>,</mo><mspace width='thinmathspace'></mspace><mi>V</mi><mo maxsize='1.2em' minsize='1.2em'>]</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>[</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace'></mspace><mo stretchy='false'>(</mo><mi>V</mi><mo>×</mo><mi>W</mi><mi>G</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>G</mi><mo maxsize='1.2em' minsize='1.2em'>]</mo></mrow><annotation encoding='application/x-tex'>
4338 \big[
4339 X \times_{\overline{W}G} W G,
4340 \,
4341 V
4342 \big]
4343 \;\simeq\;
4344 \big[
4345 X,
4346 \,
4347 (V \times W G)/G
4348 \big]
4349
4350 </annotation></semantics></math></div>
4351 <p>hence</p>
4352 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub><mi>W</mi><mi>G</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mo>•</mo><mo stretchy='false'>]</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>V</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mo>•</mo><mo stretchy='false'>]</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo stretchy='false'>(</mo><mi>V</mi><mo>×</mo><mi>W</mi><mi>G</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>G</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
4353 Hom
4354 \Big(
4355 \big( X \times_{\overline{W}G} W G \big) \times \Delta[\bullet],
4356 \,
4357 V
4358 \Big)
4359 \;\simeq\;
4360 Hom
4361 \big(
4362 X \times \Delta[\bullet],
4363 \,
4364 (V \times W G)/G
4365 \big)
4366
4367 </annotation></semantics></math></div>
4368 <p>follows from the plain adjunction and the natural isomorphism</p>
4369 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub><mi>W</mi><mi>G</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mo>•</mo><mo stretchy='false'>]</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mo>•</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub><mi>W</mi><mi>G</mi><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'>
4370 (X \times_{\overline{W}G} W G) \times \Delta[\bullet]
4371 \;\simeq\;
4372 (X \times \Delta[\bullet]) \times_{\overline{W}G} W G
4373 \,,
4374
4375 </annotation></semantics></math></div>
4376 <p>which, in turn, follows, for instance, via the <a class='existingWikiWord' href='/nlab/show/pasting+law+for+pullbacks'>pasting law</a>:</p>
4377
4378 <p>\begin{tikzcd} { { (X \times_{\overline{W}G} W G) \times \Delta[k] } \atop { \mathllap{\simeq} (X \times \Delta[k]) \times_{\overline{W}G} W G } } \ar[r] \ar[d] \ar[dr,phantom,\mbox{\tiny\rm(pb)}] &amp; X \times \Delta[k] \ar[d, \mathrm{pr}_1] \ X \times_{\overline{W}G} W G \ar[r] \ar[d] \ar[dr,phantom,\mbox{\tiny\rm(pb)}] &amp; X \ar[d] \ W G \ar[r] &amp; \overline{W}G \,. \end{tikzcd}</p>
4379
4380 <p>\end{remark}</p>
4381
4382 <h3 id='RelationToModelStructureOnPlainSimplicialSets'>Relation to the model structure on plain simplicial sets</h3>
4383
4384 <p>For <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mspace width='thinmathspace'></mspace><mo>∈</mo><mspace width='thinmathspace'></mspace><mi>Groups</mi><mo stretchy='false'>(</mo><mi>sSets</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G} \,\in\, Groups(sSets)</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a>, write <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mi>Actions</mi><mo stretchy='false'>(</mo><mi>sSets</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}Actions(sSets)</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/category'>category</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/action'>actions</a> on <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a>.</p>
4385
4386 <p>\begin{proposition}\label{CofreeAction} <strong>(underlying simplicial sets and cofree simplicial action)</strong> \linebreak The <a class='existingWikiWord' href='/nlab/show/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>undrl</mi></mrow><annotation encoding='application/x-tex'>undrl</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mi>Actions</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}Actions</annotation></semantics></math> to underlying simplicial sets is a <a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>left Quillen functor</a> from the Borel model structure (Def. \ref{BorelModelStructure}) to the <a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a>.</p>
4387
4388 <p>Its <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right adjoint</a></p>
4389 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo stretchy='false'>[</mo><mi>𝒢</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></munder><mover><mo>⟵</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>undrl</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover></munderover><mi>𝒢</mi><mi>Actions</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
4390 sSet
4391 \underoverset
4392 {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}}
4393 {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}}
4394 {\bot}
4395 \mathcal{G}Actions(sSet)
4396
4397 </annotation></semantics></math></div>
4398 <p>sends <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒳</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>\mathcal{X} \in sSet</annotation></semantics></math> to</p>
4399
4400 <ul>
4401 <li>
4402 <p>the simplicial set</p>
4403 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>𝒢</mi><mo>,</mo><mi>𝒳</mi><mo stretchy='false'>]</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mo>•</mo><mo stretchy='false'>]</mo><mo>,</mo><mi>𝒳</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>
4404 [\mathcal{G},\mathcal{X}]
4405 \;\coloneqq\;
4406 Hom_{sSet}\big( \mathcal{G} \times \Delta[\bullet], \mathcal{X}\big)
4407 \;\;\;
4408 \in
4409 sSet
4410
4411 </annotation></semantics></math></div></li>
4412
4413 <li>
4414 <p>equipped with the <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>-action</p>
4415 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo>×</mo><mo stretchy='false'>[</mo><mi>𝒢</mi><mo>,</mo><mi>𝒳</mi><mo stretchy='false'>]</mo><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>
4416 \mathcal{G} \times [\mathcal{G},\mathcal{X}]
4417 \overset{ (-) \cdot (-) }{\longrightarrow}
4418 \mathcal{G}
4419
4420 </annotation></semantics></math></div>
4421 <p>which in degree <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/function'>function</a></p>
4422 <div class='maruku-equation' id='eq:CofreeSimplicialActionComponentFunctions'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mi>𝒢</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>×</mo><mspace width='thinmathspace'></mspace><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>𝒳</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>⟶</mo><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>𝒳</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
4423
4424 Hom(\Delta[n], \mathcal{G})
4425 \,\times\,
4426 Hom
4427 \big(
4428 \mathcal{G} \times \Delta[n],
4429 \,
4430 \mathcal{X}
4431 \big)
4432 \longrightarrow
4433 Hom
4434 \big(
4435 \mathcal{G} \times \Delta[n],
4436 \,
4437 \mathcal{X}
4438 \big)
4439
4440 </annotation></semantics></math></div>
4441 <p>that sends</p>
4442 <div class='maruku-equation' id='eq:CofreeSimplicialActionInComponents'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd></mtd> <mtd><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mrow><msub><mi>g</mi> <mi>n</mi></msub></mrow></mover><mi>𝒢</mi><mo>,</mo><mspace width='thickmathspace'></mspace><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mi>ϕ</mi></mover><mi>𝒳</mi><mo>,</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr> <mtr><mtd><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo>↦</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mtd> <mtd><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><mi>diag</mi></mrow></mover><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><msub><mi>g</mi> <mi>n</mi></msub><mo>×</mo><mi>id</mi></mrow></mover><mi>𝒢</mi><mo>×</mo><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>id</mi></mrow></mover><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mi>ϕ</mi></mover><mi>𝒳</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
4443
4444 \begin{aligned}
4445 &amp;
4446 \Big(
4447 \Delta[n] \overset{g_n}{\to} \mathcal{G},
4448 \;
4449 \mathcal{G}\times \Delta[n]
4450 \overset{\phi}{\to}
4451 \mathcal{X},
4452 \Big)
4453 \\
4454 \;\;\mapsto\;\;
4455 &amp;
4456 \Big(
4457 \mathcal{G} \times \Delta[n]
4458 \overset{id \times diag}{\longrightarrow}
4459 \mathcal{G} \times \Delta[n] \times \Delta[n]
4460 \overset{ id \times g_n \times id }{\longrightarrow}
4461 \mathcal{G} \times \mathcal{G} \times \Delta[n]
4462 \overset{(-)\cdot(-) \times id}{\to}
4463 \mathcal{G} \times \Delta[n]
4464 \overset{\phi}{\to}
4465 \mathcal{X}
4466 \Big)
4467 \end{aligned}
4468
4469 </annotation></semantics></math></div></li>
4470 </ul>
4471
4472 <p>\end{proposition}</p>
4473
4474 <p>Here and in the following proof we make free use of the <a class='existingWikiWord' href='/nlab/show/Yoneda+lemma'>Yoneda lemma</a> <a class='existingWikiWord' href='/nlab/show/natural+bijection'>natural bijection</a></p>
4475 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mi>𝒮</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msub><mi>𝒮</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>
4476 Hom_{sSet}(\Delta[n], \mathcal{S}) \;\simeq\; \mathcal{S}_n
4477
4478 </annotation></semantics></math></div>
4479 <p>for any <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial set</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> and for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi><mover><mo>↪</mo><mi>y</mi></mover><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>\Delta[n] \in \Delta \overset{y}{\hookrightarrow} sSet</annotation></semantics></math> the simplicial <a class='existingWikiWord' href='/nlab/show/simplex'>n-simplex</a>.</p>
4480
4481 <p>\begin{proof}</p>
4482
4483 <p>We already know from Def. \ref{BorelModelStructure} that <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>underl</mi></mrow><annotation encoding='application/x-tex'>underl</annotation></semantics></math> preserves all <a class='existingWikiWord' href='/nlab/show/weak+equivalence'>weak equivalences</a> and from Prop. \ref{CofibrationsOfSimplicialActions} that it preserves all <a class='existingWikiWord' href='/nlab/show/cofibration'>cofibrations</a>. Therefore it is a <a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>left Quillen functor</a> as soon as it is a <a class='existingWikiWord' href='/nlab/show/left+adjoint'>left adjoint</a> at all.</p>
4484
4485 <p>The idea of the existence of the <a class='existingWikiWord' href='/nlab/show/free+functor'>cofree</a> <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right adjoint</a> to <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>undrl</mi></mrow><annotation encoding='application/x-tex'>undrl</annotation></semantics></math> is familiar from <a class='existingWikiWord' href='/nlab/show/topological+G-space'>topological G-spaces</a> (see the section on <a href='topological+G-space#CoinducedActions'>coinduced actions</a> there), where it can be easily expressed point-wise in <a class='existingWikiWord' href='/nlab/show/general+topology'>point-set topology</a>. The formula <a class='maruku-eqref' href='#eq:CofreeSimplicialActionInComponents'>(3)</a> adapts this idea to simplicial sets. Its form makes manifest that this gives a simplicial homomorphism, and with this the adjointness follows the usual logic by focusing on the image of the non-degenerate top-degree cell in <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta[n]</annotation></semantics></math>:</p>
4486
4487 <p>To check that <a class='maruku-eqref' href='#eq:CofreeSimplicialActionInComponents'>(3)</a> really gives the right adjoint, it is sufficient to check the corresponding <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a>, hence to check for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo>∈</mo><mi>𝒢</mi><mi>Actions</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{P} \in \mathcal{G}Actions(sSet)</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒳</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>\mathcal{X} \in sSet</annotation></semantics></math>, that we have a <a class='existingWikiWord' href='/nlab/show/natural+bijection'>natural bijection</a> of <a class='existingWikiWord' href='/nlab/show/hom-set'>hom-sets</a> of the form</p>
4488 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>{</mo><mi>𝒫</mi><mover><mo>⟶</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><msub><mi>ϕ</mi> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mo stretchy='false'>[</mo><mi>𝒢</mi><mo>,</mo><mi>𝒳</mi><mo stretchy='false'>]</mo><mo maxsize='1.2em' minsize='1.2em'>}</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mover><mo>↔</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mover><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><mo>˜</mo></mover><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>{</mo><mi>undrl</mi><mo stretchy='false'>(</mo><mi>𝒫</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><msub><mover><mi>ϕ</mi><mo>˜</mo></mover> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mi>𝒳</mi><mo maxsize='1.2em' minsize='1.2em'>}</mo><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
4489 \big\{
4490 \mathcal{P}
4491 \overset{\;\;\phi_{(-)}\;\;}{\longrightarrow}
4492 [\mathcal{G}, \mathcal{X}]
4493 \big\}
4494 \;\;\;\overset{ \;\; \widetilde{(-)} \;\; }{\leftrightarrow}\;\;\;
4495 \big\{
4496 undrl(\mathcal{P})
4497 \overset{\;\; {\widetilde \phi}_{(-)} \;\; }{\longrightarrow}
4498 \mathcal{X}
4499 \big\}
4500 \,.
4501
4502 </annotation></semantics></math></div>
4503 <p>So given</p>
4504 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msub><mi>p</mi> <mi>n</mi></msub><mo>↦</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>ϕ</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>𝒢</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>𝒳</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
4505 \phi_{(-)}
4506 \;\colon\;
4507 p_n
4508 \mapsto
4509 \big(
4510 \phi_{p_n}
4511 \;\colon\;
4512 \mathcal{G} \times \Delta[n] \to \mathcal{X}
4513 \big)
4514
4515 </annotation></semantics></math></div>
4516 <p>on the left, define</p>
4517 <div class='maruku-equation' id='eq:AdjunctOfHomomorphismToCofreeSimplicialAction'><span class='maruku-eq-number'>(4)</span><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>ϕ</mi><mo>˜</mo></mover> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msub><mi>p</mi> <mi>n</mi></msub><mo>↦</mo><msub><mi>ϕ</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>∈</mo><mspace width='thickmathspace'></mspace><msub><mi>𝒳</mi> <mi>n</mi></msub><mspace width='thinmathspace'></mspace><mo>,</mo></mrow><annotation encoding='application/x-tex'>
4518
4519 \widetilde \phi_{(-)}
4520 \;\colon\;
4521 p_n
4522 \mapsto
4523 \phi_{p_n}(e_n, \sigma_n)
4524 \;\in\;
4525 \mathcal{X}_n
4526 \,,
4527
4528 </annotation></semantics></math></div>
4529 <p>where <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>e</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>𝒢</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>e_n \in \mathcal{G}_n</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/identity+element'>neutral element</a> in degree <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> and where <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mi>n</mi></msub><mo>∈</mo><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><msub><mo stretchy='false'>)</mo> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\sigma_n \in (\Delta[n])_n</annotation></semantics></math> denotes the unique non-degenerate element <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-cell in the <a class='existingWikiWord' href='/nlab/show/simplex'>n-simplex</a>.</p>
4530
4531 <p>It is clear that this is a <a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a> in <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. We need to show that <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>ϕ</mi><mo>˜</mo></mover> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub><mo lspace='verythinmathspace'>:</mo><mi>undrl</mi><mo stretchy='false'>(</mo><mi>P</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>{\widetilde \phi}_{(-)} \colon undrl(P) \to X</annotation></semantics></math> uniquely determines all of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\phi_{(-)}</annotation></semantics></math>.</p>
4532
4533 <p>To that end, observe for any <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>𝒢</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>g_n \in \mathcal{G}_n</annotation></semantics></math> the following sequence of identifications:</p>
4534 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>ϕ</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mo stretchy='false'>(</mo><msub><mi>g</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><msub><mi>ϕ</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>n</mi></msub><mo>⋅</mo><msub><mi>g</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>g</mi> <mi>n</mi></msub><mo>⋅</mo><msub><mi>ϕ</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><msub><mi>ϕ</mi> <mrow><msub><mi>g</mi> <mi>n</mi></msub><mo>⋅</mo><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>=</mo><mspace width='thickmathspace'></mspace><msub><mover><mi>ϕ</mi><mo>˜</mo></mover> <mrow><msub><mi>g</mi> <mi>n</mi></msub><mo>⋅</mo><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
4535 \begin{aligned}
4536 \phi_{p_n}(g_n, \sigma_n)
4537 &amp; \;=\;
4538 \phi_{p_n}( e_n \cdot g_n, \sigma_n )
4539 \\
4540 &amp; \;=\;
4541 \big(
4542 g_n \cdot \phi_{p_n}
4543 \big)
4544 ( e_n, \sigma_n )
4545 \\
4546 &amp; \;=\;
4547 \phi_{ g_n \cdot p_n }
4548 (e_n, \sigma_n)
4549 \\
4550 &amp; \;=\;
4551 {\widetilde \phi}_{g_n \cdot p_n}
4552 \end{aligned}
4553
4554 </annotation></semantics></math></div>
4555 <p>Here:</p>
4556
4557 <ul>
4558 <li>
4559 <p>the first step is the unit law in the component group <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_n</annotation></semantics></math>;</p>
4560 </li>
4561
4562 <li>
4563 <p>the second step uses the definition <a class='maruku-eqref' href='#eq:CofreeSimplicialActionInComponents'>(3)</a> of the cofree action;</p>
4564 </li>
4565
4566 <li>
4567 <p>the third step is the assumption that <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\phi_{(-)}</annotation></semantics></math> is a homomorphism of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>\mathcal{G}</annotation></semantics></math>-actions (<a class='existingWikiWord' href='/nlab/show/equivariant'>equivariance</a>);</p>
4568 </li>
4569
4570 <li>
4571 <p>the fourth step is the definition <a class='maruku-eqref' href='#eq:AdjunctOfHomomorphismToCofreeSimplicialAction'>(4)</a>.</p>
4572 </li>
4573 </ul>
4574
4575 <p>These identifications show that <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\phi_{(-)}</annotation></semantics></math> is uniquely determined by <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><msub><mover><mi>ϕ</mi><mo>˜</mo></mover> <mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub></mrow></mrow><annotation encoding='application/x-tex'>{\widetilde \phi_{(-)}}</annotation></semantics></math>, and vice versa.</p>
4576
4577 <p>\end{proof}</p>
4578
4579 <p>\begin{example}\label{BZActionOnInertiaGroupoid} <strong>(<math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}\mathbb{Z}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/infinity-action'>2-action</a> on <a class='existingWikiWord' href='/nlab/show/inertia+orbifold'>inertia groupoid</a>)</strong> \linebreak Let</p>
4580
4581 <ul>
4582 <li>
4583 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Groups</mi><mo stretchy='false'>(</mo><mi>Sets</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G \in Groups(Sets)</annotation></semantics></math></p>
4584
4585 <p>be a <a class='existingWikiWord' href='/nlab/show/discrete+group'>discrete group</a>,</p>
4586 </li>
4587
4588 <li>
4589 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>G</mi><mi>Actions</mi><mo stretchy='false'>(</mo><mi>Sets</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \in G Actions(Sets)</annotation></semantics></math></p>
4590
4591 <p>be a <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/action'>action</a>,</p>
4592 </li>
4593
4594 <li>
4595 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒳</mi><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>X</mi><mo>⫽</mo><mi>G</mi><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>N</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>G</mi><mo>⇉</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>=</mo><mspace width='thinmathspace'></mspace><mi>X</mi><mo>×</mo><msup><mi>G</mi> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>\mathcal{X} \;\coloneqq\; X \sslash G \;\coloneqq\; N( X \times G \rightrightarrows X ) \,=\, X \times G^{\times^\bullet} \in sSet</annotation></semantics></math></p>
4596
4597 <p>the <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial set</a> which is the <a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a> of its <a class='existingWikiWord' href='/nlab/show/action+groupoid'>action groupoid</a> (a model for its <a class='existingWikiWord' href='/nlab/show/homotopy+quotient'>homotopy quotient</a>),</p>
4598 </li>
4599
4600 <li>
4601 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mspace width='thinmathspace'></mspace><mo>≔</mo><mspace width='thinmathspace'></mspace><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ℤ</mi><mspace width='thinmathspace'></mspace><mo>≔</mo><mspace width='thinmathspace'></mspace><mi>N</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo>⇉</mo><mo>*</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace'></mspace><mo>≔</mo><mspace width='thinmathspace'></mspace><msup><mi>ℤ</mi> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup><mspace width='thinmathspace'></mspace><mo>∈</mo><mspace width='thinmathspace'></mspace><mi>Groups</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G} \,\coloneqq\, \mathbf{B}\mathbb{Z} \,\coloneqq\, N(\mathbb{Z} \rightrightarrows \ast) \,\coloneqq\, \mathbb{Z}^{\times^\bullet} \,\in\, Groups(sSet)</annotation></semantics></math></p>
4602
4603 <p>the <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial group</a> which is the <a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a> of the <a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a> that is the <a class='existingWikiWord' href='/nlab/show/delooping+groupoid'>delooping groupoid</a> of the additive group of <a class='existingWikiWord' href='/nlab/show/integer'>integers</a>.</p>
4604 </li>
4605 </ul>
4606
4607 <p>Then the <a class='existingWikiWord' href='/nlab/show/functor+category'>functor groupoid</a></p>
4608 <div class='maruku-equation' id='eq:InertiaGroupoidAsFunctorGroupoidOutOfBZ'><span class='maruku-eq-number'>(5)</span><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>Λ</mi><mo stretchy='false'>(</mo><mi>X</mi><mspace width='negativethinmathspace'></mspace><mo>⫽</mo><mspace width='negativethinmathspace'></mspace><mi>G</mi><mo stretchy='false'>)</mo></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>[</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ℤ</mi><mo>,</mo><mi>X</mi><mspace width='negativethinmathspace'></mspace><mo>⫽</mo><mspace width='negativethinmathspace'></mspace><mi>G</mi><mo maxsize='1.2em' minsize='1.2em'>]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>Func</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mi>ℤ</mi><mo>⇉</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>G</mi><mo>⇉</mo><mi>X</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><munder><mo>←</mo><mrow><mo>∈</mo><mi mathvariant='normal'>W</mi></mrow></munder><mspace width='thickmathspace'></mspace><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mrow><mo stretchy='false'>[</mo><mi>g</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>ConjCl</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow></munder><mo maxsize='1.8em' minsize='1.8em'>(</mo><msup><mi>X</mi> <mi>g</mi></msup><mspace width='negativethinmathspace'></mspace><mo>⫽</mo><mspace width='negativethinmathspace'></mspace><msub><mi>C</mi> <mi>g</mi></msub><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
4609
4610 \begin{aligned}
4611 \Lambda(X \!\sslash\! G)
4612 &amp; \;\coloneqq\;
4613 \big[
4614 \mathbf{B}\mathbb{Z}, X \!\sslash\! G
4615 \big]
4616 \\
4617 &amp;
4618 \;\simeq\;
4619 Func
4620 \big(
4621 (\mathbb{Z} \rightrightarrows \ast),
4622 \,
4623 (X \times G \rightrightarrows X)
4624 \big)
4625 \\
4626 &amp; \;\underset{\in \mathrm{W}}{\leftarrow}\;
4627 \underset{
4628 [g] \in ConjCl(G)
4629 }{\coprod}
4630 \Big(
4631 X^{g} \!\sslash\! C_g
4632 \Big)
4633 \end{aligned}
4634
4635 </annotation></semantics></math></div>
4636 <p>is known as the <em><a class='existingWikiWord' href='/nlab/show/inertia+orbifold'>inertia groupoid</a></em> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mspace width='negativethinmathspace'></mspace><mo>⫽</mo><mspace width='negativethinmathspace'></mspace><mi>G</mi></mrow><annotation encoding='application/x-tex'>X \!\sslash\! G</annotation></semantics></math>. Here</p>
4637 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ConjCla</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>G</mi><msub><mo stretchy='false'>/</mo> <mi>ad</mi></msub><mi>G</mi><mspace width='thinmathspace'></mspace><mo>,</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><msub><mi>C</mi> <mi>g</mi></msub><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>{</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mspace width='thinmathspace'></mspace><mrow><mo>|</mo><mspace width='thinmathspace'></mspace><mi>h</mi><mo>⋅</mo><mi>g</mi><mo>=</mo><mi>g</mi><mo>⋅</mo><mi>h</mi></mrow><mo maxsize='1.2em' minsize='1.2em'>}</mo></mrow><annotation encoding='application/x-tex'>
4638 ConjCla(G)
4639 \;\coloneqq\;
4640 G/_{ad} G
4641 \,,
4642 \;\;\;\;\;\;\;\;\;\;\;
4643 C_g
4644 \;\coloneqq\;
4645 \big\{
4646 h \in G
4647 \,\left\vert\,
4648 h \cdot g = g \cdot h
4649 \right.
4650 \big\}
4651
4652 </annotation></semantics></math></div>
4653 <p>denotes, respectively, the set of <a class='existingWikiWord' href='/nlab/show/conjugacy+class'>conjugacy classes</a> of elements of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, and the <a class='existingWikiWord' href='/nlab/show/centralizer'>centralizer</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>g</mi><mo stretchy='false'>}</mo><mo>⊂</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>\{g\} \subset G</annotation></semantics></math> – this data serves to express the <a class='existingWikiWord' href='/nlab/show/equivalence+of+categories'>equivalent</a> <a class='existingWikiWord' href='/nlab/show/skeleton'>skeleton</a> of the inertia groupoid in the last line of <a class='maruku-eqref' href='#eq:InertiaGroupoidAsFunctorGroupoidOutOfBZ'>(5)</a>.</p>
4654
4655 <p>Now, by Prop. \ref{CofreeAction} the inertia groupoid <a class='maruku-eqref' href='#eq:InertiaGroupoidAsFunctorGroupoidOutOfBZ'>(5)</a> carries a canonical <a class='existingWikiWord' href='/nlab/show/infinity-action'>2-action</a> of the <a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a> <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}\mathbb{Z}</annotation></semantics></math>:</p>
4656
4657 <p>By the formula <a class='maruku-eqref' href='#eq:CofreeSimplicialActionInComponents'>(3)</a>, for <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{Z}</annotation></semantics></math> the 2-group element in degree 1</p>
4658 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathcolor='purple'><mi>n</mi></mstyle><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⟶</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>
4659 {\color{purple}n}
4660 \;\colon\;
4661 \Delta[1]
4662 \longrightarrow
4663 \mathbf{B}G
4664
4665 </annotation></semantics></math></div>
4666 <p>acts on the morphisms</p>
4667 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mi>h</mi></mover><mo stretchy='false'>(</mo><mi>h</mi><mo>⋅</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mo>∈</mo><mspace width='thickmathspace'></mspace><mi>Λ</mi><mo stretchy='false'>(</mo><mi>X</mi><mspace width='negativethinmathspace'></mspace><mo>⫽</mo><mspace width='negativethinmathspace'></mspace><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
4668 (x,g) \overset{h}{\longrightarrow} (h\cdot x, g)
4669 \;\;\;
4670 \in
4671 \;
4672 \Lambda(X \!\sslash\! G)
4673
4674 </annotation></semantics></math></div>
4675 <p>of the inertia groupoid as follows (recall the nature of <a class='existingWikiWord' href='/nlab/show/product+of+simplices'>products of simplices</a>):</p>
4676
4677 <p><img src='https://ncatlab.org/nlab/files/BZActionOnInertiaGroupoid20210624.jpg' width='800'/></p>
4678
4679 <p>\end{example}</p>
4680
4681 <h3 id='relation_to_the_fine_model_structure_of_equivariant_homotopy_theory'>Relation to the fine model structure of equivariant homotopy theory</h3>
4682
4683 <p>The <a class='existingWikiWord' href='/nlab/show/identity+functor'>identity functor</a> gives a <a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>Quillen adjunction</a> between the Borel model structure and <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant homotopy theory</a> (<a href='#Guillou'>Guillou, section 5</a>).</p>
4684
4685 <p>The left adjoint is</p>
4686 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>=</mo><mi>id</mi><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msub><mi>G</mi> <mo>•</mo></msub><msub><mi>Act</mi> <mi>coarse</mi></msub><mo>⟶</mo><msub><mi>G</mi> <mo>•</mo></msub><msub><mi>Act</mi> <mi>fine</mi></msub></mrow><annotation encoding='application/x-tex'>
4687 L = id
4688 \;\colon\;
4689 G_\bullet Act_{coarse}
4690 \longrightarrow
4691 G_\bullet Act_{fine}
4692
4693 </annotation></semantics></math></div>
4694 <p>from the Borel model structure to the genuine <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant homotopy theory</a>.</p>
4695
4696 <p>Because:</p>
4697
4698 <p>First of all, by (<a href='#Guillou'>Guillou, theorem 3.12, example 4.2</a>) <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>sSet</mi> <mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow></msup></mrow><annotation encoding='application/x-tex'>sSet^{\mathbf{B}G_\bullet}</annotation></semantics></math> does carry a fine model structure. By (<a href='#Guillou'>Guillou, last line of page 3</a>) the fibrations and weak equivalences here are those maps which are ordinary fibrations and weak equivalences, respectively, on <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/fixed+point'>fixed point</a> simplicial sets, for all subgroups <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. This includes in particular the trivial subgroup and hence the identity functor</p>
4699 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>=</mo><mi>id</mi><mspace width='thickmathspace'></mspace><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace'></mspace><msub><mi>G</mi> <mo>•</mo></msub><msub><mi>Act</mi> <mi>fine</mi></msub><mo>⟶</mo><msub><mi>G</mi> <mo>•</mo></msub><msub><mi>Act</mi> <mi>coarse</mi></msub></mrow><annotation encoding='application/x-tex'>
4700 R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{coarse}
4701
4702 </annotation></semantics></math></div>
4703 <p>is right Quillen.</p>
4704
4705 <h3 id='GeneralizationToSimplicialPresheaves'>Generalization to simplicial presheaves</h3>
4706
4707 <p>Since the <a class='existingWikiWord' href='/nlab/show/simplicial+classifying+space'>universal simplicial principal complex</a>-construction is <a class='existingWikiWord' href='/nlab/show/functor'>functorial</a></p>
4708 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SimplicialGroups</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>W</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mi>SimplicialSets</mi></mrow><annotation encoding='application/x-tex'>
4709 SimplicialGroups
4710 \xrightarrow{\;\; W \;\;}
4711 SimplicialSets
4712
4713 </annotation></semantics></math></div>
4714 <p>with <a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformations</a></p>
4715 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>i</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mi>W</mi><mi>𝒢</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>p</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>
4716 \mathcal{G}
4717 \xrightarrow{\;\; i \;\;}
4718 W\mathcal{G}
4719 \xrightarrow{\;\; p \;\;}
4720 \overline{W}\mathcal{G}
4721
4722 </annotation></semantics></math></div>
4723 <p>the pair of <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functors</a> <a class='maruku-eqref' href='#eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace'>(1)</a> extends to <a class='existingWikiWord' href='/nlab/show/presheaf'>presheaves</a>:</p>
4724
4725 <p>\begin{prop} For <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/small+category'>small</a> <a class='existingWikiWord' href='/nlab/show/simplicially+enriched+category'>sSet-category</a> with</p>
4726 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>sSetCat</mi><mo stretchy='false'>(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mspace width='thinmathspace'></mspace><mi>sSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
4727 sPSh(\mathcal{C})
4728 \;\coloneqq\;
4729 sSetCat( \mathcal{C}^{op}, \, sSet )
4730
4731 </annotation></semantics></math></div>
4732 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>simplicial presheaves</a>, and for</p>
4733 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mspace width='thickmathspace'></mspace><mo>∈</mo><mspace width='thickmathspace'></mspace><mi>Groups</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
4734 \underline{\mathcal{G}}
4735 \;\in\;
4736 Groups
4737 \big(
4738 sPSh(\mathcal{C})
4739 \big)
4740
4741 </annotation></semantics></math></div>
4742 <p>a <a class='existingWikiWord' href='/nlab/show/group+object'>group object</a> <a class='existingWikiWord' href='/nlab/show/internalization'>internal to</a> <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>SimplicialPresheaves</a> with</p>
4743 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
4744 \underline{\mathcal{G}}
4745 Acts
4746 \big(
4747 sPSh(\mathcal{C})
4748 \big)
4749
4750 </annotation></semantics></math></div>
4751 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show/module+object'>action objects</a> <a class='existingWikiWord' href='/nlab/show/internalization'>internal to</a> <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>SimplicialPresheaves</a></p>
4752
4753 <p>we have an <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint pair</a></p>
4754 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.2em' minsize='1.2em'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></mover></munderover><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub></mrow><annotation encoding='application/x-tex'>
4755 \underline{\mathcal{G}}
4756 Acts
4757 \big(
4758 sPSh(\mathcal{C})
4759 \big)
4760 \underoverset
4761 {
4762 \underset{
4763 \big(
4764 (-) \times W\underline{\mathcal{G}}
4765 \big)
4766 \big/
4767 \underline{\mathcal{G}}
4768 }
4769 {\longrightarrow}}
4770 {
4771 \overset{
4772 (-)
4773 \times_{\overline{W}\underline{\mathcal{G}}}
4774 W\underline{\mathcal{G}}
4775 }{\longleftarrow}
4776 }
4777 {\bot}
4778 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}}
4779
4780 </annotation></semantics></math></div>
4781 <p>\end{prop} \begin{proof}</p>
4782
4783 <p>The required <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> is the composite of the following sequence of <a class='existingWikiWord' href='/nlab/show/natural+bijection'>natural bijections</a>:</p>
4784 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo stretchy='false'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mi>p</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><msup><mo>∫</mo> <mi>c</mi></msup><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mrow><mo>(</mo><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><msub><mi>Hom</mi> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.2em' minsize='1.2em'>/</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mrow><mo>(</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><munder><mo>×</mo><mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></munder><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>𝒢</mi><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><munder><mo>×</mo><mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>Y</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
4785 \begin{aligned}
4786 Hom
4787 \Big(
4788 (\underline{X},p),
4789 \,
4790 \big(
4791 \underline{Y} \times W\underline{\mathcal{G}}
4792 \big) / \underline{\mathcal{G}}
4793 \Big)
4794 &amp;
4795 \;\simeq\;
4796 Hom
4797 \Big(
4798 \underline{X},
4799 \,
4800 \big(
4801 \underline{Y} \times W\underline{\mathcal{G}}
4802 \big) / \underline{\mathcal{G}}
4803 \Big)
4804 \underset{
4805 Hom
4806 \Big(
4807 \underline{X},
4808 \,
4809 \overline{W} \underline{\mathcal{G}}
4810 \Big)
4811 }{\times}
4812 \{p\}
4813 \\
4814 &amp; \;\simeq\;
4815 \int^c
4816 Hom
4817 \Big(
4818 \underline{X}(c),
4819 \,
4820 \big(
4821 \underline{Y}(c) \times W\underline{\mathcal{G}(c)}
4822 \big) / \underline{\mathcal{G}}(c)
4823 \Big)
4824 \underset{
4825 \int^c
4826 Hom
4827 \Big(
4828 \underline{X}(c),
4829 \,
4830 \overline{W} \underline{\mathcal{G}}(c)
4831 \Big)
4832 }{\times}
4833 \{p\}
4834 \\
4835 &amp; \;\simeq\;
4836 \int^c
4837 \left(
4838 Hom
4839 \Big(
4840 \underline{X}(c),
4841 \,
4842 \big(
4843 \underline{Y}(c) \times W\underline{\mathcal{G}}(c)
4844 \big) / \underline{\mathcal{G}}(c)
4845 \Big)
4846 \underset{
4847 Hom
4848 \Big(
4849 \underline{X}(c),
4850 \,
4851 \overline{W} \underline{\mathcal{G}}(c)
4852 \Big)
4853 }{\times}
4854 \{p(c)\}
4855 \right)
4856 \\
4857 &amp; \;\simeq\;
4858 \int^c
4859 Hom_{/\overline{W}\underline{\mathcal{G}}(c)}
4860 \Big(
4861 \big( \underline{X}(c), p(c)\big),
4862 \,
4863 \big(
4864 \underline{Y}(c) \times \overline{W} \underline{\mathcal{G}}(c)
4865 \big)\big/ \mathcal{G}(c)
4866 \Big)
4867 \\
4868 &amp; \;\simeq\;
4869 \int^c
4870 \left(
4871 \underline{\mathcal{G}}(c)
4872 Acts(sSet)
4873 \big(
4874 \underline{X}(c)
4875 \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times}
4876 W \underline{\mathcal{G}}(c),
4877 \,
4878 \underline{Y}(c)
4879 \big)
4880 \right)
4881 \\
4882 &amp; \;\simeq\;
4883 \mathcal{G}Acts(sPSh(\mathcal{C}))
4884 \big(
4885 \underline{X}
4886 \underset{\overline{W}\underline{\mathcal{G}}}{\times}
4887 W \underline{\mathcal{G}},
4888 \,
4889 \underline{Y}
4890 \big)
4891 \end{aligned}
4892
4893 </annotation></semantics></math></div>
4894 <p>Here:</p>
4895
4896 <ul>
4897 <li>
4898 <p>the first step is the characterization of hom-sets of a <a class='existingWikiWord' href='/nlab/show/over+category'>slice category</a> as a <a class='existingWikiWord' href='/nlab/show/fiber'>fiber</a> of the <a class='existingWikiWord' href='/nlab/show/hom-set'>hom-sets</a> of the underlying category;</p>
4899 </li>
4900
4901 <li>
4902 <p>the second step is the description of the hom-set of a <a class='existingWikiWord' href='/nlab/show/functor+category'>functor category</a> as an <a class='existingWikiWord' href='/nlab/show/end'>end</a> of object-wise hom-sets;</p>
4903 </li>
4904
4905 <li>
4906 <p>the third step uses that <a class='existingWikiWord' href='/nlab/show/end'>ends</a> are <a class='existingWikiWord' href='/nlab/show/limit'>limits</a> and <a class='existingWikiWord' href='/nlab/show/limits+commute+with+limits'>hence commute</a> the the <a class='existingWikiWord' href='/nlab/show/pullback'>fiber product</a>;</p>
4907 </li>
4908
4909 <li>
4910 <p>the fourth step recognizes again, now object-wise, the hom-set in a <a class='existingWikiWord' href='/nlab/show/over+category'>slice category</a>;</p>
4911 </li>
4912
4913 <li>
4914 <p>the fifth step is objectwise the <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> of <a class='maruku-eqref' href='#eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace'>(1)</a>;</p>
4915 </li>
4916
4917 <li>
4918 <p>the sixth step recognizes again the <a class='existingWikiWord' href='/nlab/show/end'>end</a> as computing the hom-set in (a subcategory of) a functor category:</p>
4919 </li>
4920 </ul>
4921 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>B</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>𝒢</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd></mtr> <mtr><mtd><mi>𝒢</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
4922 \array{
4923 \underline{\mathcal{G}}Acts
4924 \big(
4925 \underline{A}, \, \underline{B}
4926 \big)
4927 &amp;\longrightarrow&amp;
4928 \mathcal{G}(c_1)Acts
4929 \big(
4930 \underline{A}(c_1), \, \underline{B}(c_1)
4931 \big)
4932 \\
4933 \big\downarrow
4934 &amp;&amp;
4935 \big\downarrow
4936 \\
4937 \mathcal{G}(c_2)Acts
4938 \big(
4939 \underline{A}(c_2), \, \underline{B}(c_2)
4940 \big)
4941 &amp;\longrightarrow&amp;
4942 Hom
4943 \big(
4944 \underline{A}(c_1), \, \underline{B}(c_2)
4945 \big)
4946 }
4947
4948 </annotation></semantics></math></div>
4949 <p>\end{proof}</p>
4950
4951 <h2 id='references'>References</h2>
4952
4953 <p>The model structure, the characterization of its cofibrations, and its equivalence to the <a class='existingWikiWord' href='/nlab/show/model+structure+on+an+over+category'>slice model structure</a> of <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d98884c376eeb359bc6befc3ec409_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>W</mi><mo stretchy='false'>¯</mo></mover><mi>G</mi></mrow><annotation encoding='application/x-tex'>\bar W G</annotation></semantics></math> is due to</p>
4954
4955 <ul>
4956 <li id='DDK80'><a class='existingWikiWord' href='/nlab/show/Emmanuel+Dror+Farjoun'>Emmanuel Dror</a>, <a class='existingWikiWord' href='/nlab/show/William+Dwyer'>William Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/Daniel+Kan'>Daniel Kan</a>, <em>Equivariant maps which are self homotopy equivalences</em>, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (<a href='http://www.jstor.org/stable/2043448'>jstor:2043448</a>)</li>
4957 </ul>
4958
4959 <p>This Quillen equivalence also mentioned as:</p>
4960
4961 <ul>
4962 <li id='Dwyer2008'><a class='existingWikiWord' href='/nlab/show/William+Dwyer'>William Dwyer</a>, Exercise 4.2 in: <em>Homotopy theory of classifying spaces</em>, Lecture notes, Copenhagen 2008, (<a href='http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf' title='pdf'>pdf</a>)</li>
4963 </ul>
4964
4965 <p>Discussion in relation to the “fine” model structure of <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant homotopy theory</a> which appears in <a class='existingWikiWord' href='/nlab/show/Elmendorf%27s+theorem'>Elmendorf&#39;s theorem</a> is in</p>
4966
4967 <ul>
4968 <li id='Guillou'><a class='existingWikiWord' href='/nlab/show/Bert+Guillou'>Bert Guillou</a>, <em>A short note on models for equivariant homotopy theory</em>, 2006 (<a href='http://www.math.uiuc.edu/~bertg/EquivModels.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/GuillouModelsForEquivariantHomotopyTheory.pdf' title='pdf'>pdf</a>)</li>
4969 </ul>
4970
4971 <p>Textbook account of (just) the Borel model structure:</p>
4972
4973 <ul>
4974 <li id='GoerssJardine09'><a class='existingWikiWord' href='/nlab/show/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, Section V.2 of: <em><a class='existingWikiWord' href='/nlab/show/Simplicial+homotopy+theory'>Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (<a href='https://link.springer.com/book/10.1007/978-3-0346-0189-4'>doi:10.1007/978-3-0346-0189-4</a>, <a href='http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/'>webpage</a>)</li>
4975 </ul>
4976
4977 <p>Discussion with the model of <a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-groups</a> by <a class='existingWikiWord' href='/nlab/show/simplicial+group'>simplicial groups</a> replaced by groupal <a class='existingWikiWord' href='/nlab/show/Segal+space'>Segal spaces</a> is in</p>
4978
4979 <ul>
4980 <li><a class='existingWikiWord' href='/nlab/show/Matan+Prasma'>Matan Prasma</a>, <em>Segal Group Actions</em> (<a href='http://arxiv.org/abs/1311.4749'>arXiv:1311.4749</a>)</li>
4981 </ul>
4982
4983 <p>Discussion of a <a class='existingWikiWord' href='/nlab/show/global+equivariant+homotopy+theory'>globalized</a> model structure for actions of all simplicial groups is in</p>
4984
4985 <ul>
4986 <li><a class='existingWikiWord' href='/nlab/show/Yonatan+Harpaz'>Yonatan Harpaz</a>, <a class='existingWikiWord' href='/nlab/show/Matan+Prasma'>Matan Prasma</a>, section 6.2 of <em>The Grothendieck construction for model categories</em> (<a href='http://arxiv.org/abs/1404.1852'>arXiv:1404.1852</a>)</li>
4987 </ul>
4988
4989 <p>
4990 </p> </div>
4991 </content>
4992 </entry>
4993 <entry>
4994 <title type="html">Sandbox</title>
4995 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Sandbox"/>
4996 <updated>2021-07-01T16:14:56Z</updated>
4997 <published>2009-07-07T06:11:26Z</published>
4998 <id>tag:ncatlab.org,2009-07-07:nLab,Sandbox</id>
4999 <author>
5000 <name>Urs Schreiber</name>
5001 </author>
5002 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Sandbox">
5003 <div xmlns="http://www.w3.org/1999/xhtml">
5004 <h3 id='GeneralizationToSimplicialPresheaves'>Generalization to simplicial presheaves</h3>
5005
5006 <p>Since the <a class='existingWikiWord' href='/nlab/show/simplicial+classifying+space'>universal simplicial principal complex</a>-construction is <a class='existingWikiWord' href='/nlab/show/functor'>functorial</a></p>
5007 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SimplicialGroups</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>W</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mi>SimplicialSets</mi></mrow><annotation encoding='application/x-tex'>
5008 SimplicialGroups
5009 \xrightarrow{\;\; W \;\;}
5010 SimplicialSets
5011
5012 </annotation></semantics></math></div>
5013 <p>with <a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformations</a></p>
5014 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>i</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mi>W</mi><mi>𝒢</mi><mover><mo>→</mo><mrow><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace><mi>p</mi><mspace width='thickmathspace'></mspace><mspace width='thickmathspace'></mspace></mrow></mover><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow><annotation encoding='application/x-tex'>
5015 \mathcal{G}
5016 \xrightarrow{\;\; i \;\;}
5017 W\mathcal{G}
5018 \xrightarrow{\;\; p \;\;}
5019 \overline{W}\mathcal{G}
5020
5021 </annotation></semantics></math></div>
5022 <p>the pair of <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functors</a> (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace) extends to <a class='existingWikiWord' href='/nlab/show/presheaf'>presheaves</a>:</p>
5023
5024 <p>\begin{prop} For <math class='maruku-mathml' display='inline' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/small+category'>small</a> <a class='existingWikiWord' href='/nlab/show/simplicially+enriched+category'>sSet-category</a> with</p>
5025 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>≔</mo><mspace width='thickmathspace'></mspace><mi>sSetCat</mi><mo stretchy='false'>(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mspace width='thinmathspace'></mspace><mi>sSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>
5026 sPSh(\mathcal{C})
5027 \;\coloneqq\;
5028 sSetCat( \mathcal{C}^{op}, \, sSet )
5029
5030 </annotation></semantics></math></div>
5031 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>simplicial presheaves</a>, and for</p>
5032 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mspace width='thickmathspace'></mspace><mo>∈</mo><mspace width='thickmathspace'></mspace><mi>Groups</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
5033 \underline{\mathcal{G}}
5034 \;\in\;
5035 Groups
5036 \big(
5037 sPSh(\mathcal{C})
5038 \big)
5039
5040 </annotation></semantics></math></div>
5041 <p>a <a class='existingWikiWord' href='/nlab/show/group+object'>group object</a> <a class='existingWikiWord' href='/nlab/show/internalization'>internal to</a> <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>SimplicialPresheaves</a> with</p>
5042 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>
5043 \underline{\mathcal{G}}
5044 Acts
5045 \big(
5046 sPSh(\mathcal{C})
5047 \big)
5048
5049 </annotation></semantics></math></div>
5050 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show/module+object'>action objects</a> <a class='existingWikiWord' href='/nlab/show/internalization'>internal to</a> <a class='existingWikiWord' href='/nlab/show/simplicial+presheaf'>SimplicialPresheaves</a></p>
5051
5052 <p>we have an <a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint pair</a></p>
5053 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.2em' minsize='1.2em'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></mover></munderover><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub></mrow><annotation encoding='application/x-tex'>
5054 \underline{\mathcal{G}}
5055 Acts
5056 \big(
5057 sPSh(\mathcal{C})
5058 \big)
5059 \underoverset
5060 {
5061 \underset{
5062 \big(
5063 (-) \times W\underline{\mathcal{G}}
5064 \big)
5065 \big/
5066 \underline{\mathcal{G}}
5067 }
5068 {\longrightarrow}}
5069 {
5070 \overset{
5071 (-)
5072 \times_{\overline{W}\underline{\mathcal{G}}}
5073 W\underline{\mathcal{G}}
5074 }{\longleftarrow}
5075 }
5076 {\bot}
5077 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}}
5078
5079 </annotation></semantics></math></div>
5080 <p>\end{prop} \begin{proof}</p>
5081
5082 <p>The required <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> is the composite of the following sequence of <a class='existingWikiWord' href='/nlab/show/natural+bijection'>natural bijections</a>:</p>
5083 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo stretchy='false'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mi>p</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><msup><mo>∫</mo> <mi>c</mi></msup><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mrow><mo>(</mo><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><munder><mi>Y</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><munder><mo>×</mo><mrow><mi>Hom</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>X</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow></munder><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><msub><mi>Hom</mi> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>×</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.2em' minsize='1.2em'>/</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mrow><mo>(</mo><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><munder><mo>×</mo><mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></munder><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>Y</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width='thickmathspace'></mspace><mo>≃</mo><mspace width='thickmathspace'></mspace><mi>𝒢</mi><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><munder><mo>×</mo><mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo>,</mo><mspace width='thinmathspace'></mspace><mi>Y</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
5084 \begin{aligned}
5085 Hom
5086 \Big(
5087 (\underline{X},p),
5088 \,
5089 \big(
5090 \underline{Y} \times W\underline{\mathcal{G}}
5091 \big) / \underline{\mathcal{G}}
5092 \Big)
5093 &amp;
5094 \;\simeq\;
5095 Hom
5096 \Big(
5097 \underline{X},
5098 \,
5099 \big(
5100 \underline{Y} \times W\underline{\mathcal{G}}
5101 \big) / \underline{\mathcal{G}}
5102 \Big)
5103 \underset{
5104 Hom
5105 \Big(
5106 \underline{X},
5107 \,
5108 \overline{W} \underline{\mathcal{G}}
5109 \Big)
5110 }{\times}
5111 \{p\}
5112 \\
5113 &amp; \;\simeq\;
5114 \int^c
5115 Hom
5116 \Big(
5117 \underline{X}(c),
5118 \,
5119 \big(
5120 \underline{Y}(c) \times W\underline{\mathcal{G}(c)}
5121 \big) / \underline{\mathcal{G}}(c)
5122 \Big)
5123 \underset{
5124 \int^c
5125 Hom
5126 \Big(
5127 \underline{X}(c),
5128 \,
5129 \overline{W} \underline{\mathcal{G}}(c)
5130 \Big)
5131 }{\times}
5132 \{p\}
5133 \\
5134 &amp; \;\simeq\;
5135 \int^c
5136 \left(
5137 Hom
5138 \Big(
5139 \underline{X}(c),
5140 \,
5141 \big(
5142 \underline{Y}(c) \times W\underline{\mathcal{G}(c)}
5143 \big) / \underline{\mathcal{G}}(c)
5144 \Big)
5145 \underset{
5146 Hom
5147 \Big(
5148 \underline{X}(c),
5149 \,
5150 \overline{W} \underline{\mathcal{G}}(c)
5151 \Big)
5152 }{\times}
5153 \{p(c)\}
5154 \right)
5155 \\
5156 &amp; \;\simeq\;
5157 \int^c
5158 Hom_{/\overline{W}\underline{\mathcal{G}}(c)}
5159 \Big(
5160 \big(X(c), p(c)\big),
5161 \,
5162 \big(
5163 Y(c) \times \overline{W} \mathcal{G}(c)
5164 \big)\big/ \mathcal{G}(c)
5165 \Big)
5166 \\
5167 &amp; \;\simeq\;
5168 \int^c
5169 \left(
5170 \underline{\mathcal{G}}(c)
5171 Acts(sSet)
5172 \big(
5173 X(c)
5174 \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times}
5175 W \underline{\mathcal{G}}(c),
5176 \,
5177 Y(c)
5178 \big)
5179 \right)
5180 \\
5181 &amp; \;\simeq\;
5182 \mathcal{G}Acts(sPSh(\mathcal{C}))
5183 \big(
5184 X
5185 \underset{\overline{W}\underline{\mathcal{G}}}{\times}
5186 W \underline{\mathcal{G}},
5187 \,
5188 Y
5189 \big)
5190 \end{aligned}
5191
5192 </annotation></semantics></math></div>
5193 <p>Here:</p>
5194
5195 <ul>
5196 <li>
5197 <p>the first step is the characterization of hom-sets of a <a class='existingWikiWord' href='/nlab/show/over+category'>slice category</a> as a <a class='existingWikiWord' href='/nlab/show/fiber'>fiber</a> of the <a class='existingWikiWord' href='/nlab/show/hom-set'>hom-sets</a> of the underlying category;</p>
5198 </li>
5199
5200 <li>
5201 <p>the second step is the description of the hom-set of a <a class='existingWikiWord' href='/nlab/show/functor+category'>functor category</a> as an <a class='existingWikiWord' href='/nlab/show/end'>end</a> of object-wise hom-sets;</p>
5202 </li>
5203
5204 <li>
5205 <p>the third step uses that <a class='existingWikiWord' href='/nlab/show/end'>ends</a> are <a class='existingWikiWord' href='/nlab/show/limit'>limits</a> and <a class='existingWikiWord' href='/nlab/show/limits+commute+with+limits'>hence commute</a> the the <a class='existingWikiWord' href='/nlab/show/pullback'>fiber product</a>;</p>
5206 </li>
5207
5208 <li>
5209 <p>the fourth step recognizes again, now object-wise, the hom-set in a <a class='existingWikiWord' href='/nlab/show/over+category'>slice category</a>;</p>
5210 </li>
5211
5212 <li>
5213 <p>the fifth step is objectwise the <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> of (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace);</p>
5214 </li>
5215
5216 <li>
5217 <p>the sixth step recognizes again the <a class='existingWikiWord' href='/nlab/show/end'>end</a> as computing the hom-set in (a subcategory of) a functor category:</p>
5218 </li>
5219 </ul>
5220 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>A</mi><mo>,</mo><mspace width='thinmathspace'></mspace><mi>B</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>𝒢</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>B</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd></mtr> <mtr><mtd><mi>𝒢</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>B</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Hom</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace'></mspace><mi>B</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
5221 \array{
5222 \underline{\mathcal{G}}Acts
5223 \big(
5224 A, \, B
5225 \big)
5226 &amp;\longrightarrow&amp;
5227 \mathcal{G}(c_1)Acts
5228 \big(
5229 A(c_1), \, B(c_1)
5230 \big)
5231 \\
5232 \big\downarrow
5233 &amp;&amp;
5234 \big\downarrow
5235 \\
5236 \mathcal{G}(c_2)Acts
5237 \big(
5238 A(c_2), \, B(c_2)
5239 \big)
5240 &amp;\longrightarrow&amp;
5241 Hom
5242 \big(
5243 A(c_1), \, B(c_2)
5244 \big)
5245 }
5246
5247 </annotation></semantics></math></div>
5248 <p>\end{proof}</p>
5249
5250 <p>\linebreak</p>
5251
5252 <p>\linebreak</p>
5253
5254 <p>added the observation (<a href='https://ncatlab.org/nlab/show/Borel+model+structure#GeneralizationToSimplicialPresheaves'>here</a>) that the adjunction for simplicial groups</p>
5255 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mi>Acts</mi><mo stretchy='false'>(</mo><mi>sSet</mi><mo stretchy='false'>)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><mi>𝒢</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><mi>𝒢</mi></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow></msub><mi>W</mi><mi>𝒢</mi></mrow></mover></munderover><msub><mi>sSet</mi> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>
5256 \mathcal{G} Acts(sSet)
5257 \underoverset
5258 {\underset{ \big((-) \times W \mathcal{G}\big)/\mathcal{G} }{\longrightarrow}}
5259 {\overset{ (-) \times_{\overline{W}\mathcal{G}} W \mathcal{G} }{\longleftarrow}}
5260 {\bot}
5261 sSet_{/\overline{W}\mathcal{G}}
5262
5263 </annotation></semantics></math></div>
5264 <p>generalizes to one for presheaves of simplicial groups</p>
5265 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fbe839e76a492d60225f2c47ba18f21550b3417b_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.2em' minsize='1.2em'>/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></mover></munderover><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub></mrow><annotation encoding='application/x-tex'>
5266 \underline{\mathcal{G}}
5267 Acts
5268 \big(
5269 sPSh(\mathcal{C})
5270 \big)
5271 \underoverset
5272 {
5273 \underset{
5274 \big(
5275 (-) \times W\underline{\mathcal{G}}
5276 \big)
5277 \big/
5278 \underline{\mathcal{G}}
5279 }
5280 {\longrightarrow}}
5281 {
5282 \overset{
5283 (-)
5284 \times_{\overline{W}\underline{\mathcal{G}}}
5285 W\underline{\mathcal{G}}
5286 }{\longleftarrow}
5287 }
5288 {\bot}
5289 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}}
5290
5291 </annotation></semantics></math></div> </div>
5292 </content>
5293 </entry>
5294 <entry>
5295 <title type="html">simplicial presheaf</title>
5296 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/simplicial+presheaf"/>
5297 <updated>2021-07-01T14:31:25Z</updated>
5298 <published>2009-01-29T18:34:04Z</published>
5299 <id>tag:ncatlab.org,2009-01-29:nLab,simplicial+presheaf</id>
5300 <author>
5301 <name>Urs Schreiber</name>
5302 </author>
5303 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/simplicial+presheaf">
5304 <div xmlns="http://www.w3.org/1999/xhtml">
5305 <div class='rightHandSide'>
5306 <div class='toc clickDown' tabindex='0'>
5307 <h3 id='context'>Context</h3>
5308
5309 <h4 id='homotopy_theory'>Homotopy theory</h4>
5310
5311 <div class='hide'>
5312 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a></strong></p>
5313
5314 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/directed+homotopy+theory'>directed</a>…</p>
5315
5316 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/localic+homotopy+theory'>localic</a>, …</p>
5317
5318 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong></p>
5319
5320 <p><strong>Introductions</strong></p>
5321
5322 <ul>
5323 <li>
5324 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p>
5325 </li>
5326
5327 <li>
5328 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p>
5329 </li>
5330
5331 <li>
5332 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p>
5333 </li>
5334 </ul>
5335
5336 <p><strong>Definitions</strong></p>
5337
5338 <ul>
5339 <li>
5340 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/higher+homotopy'>higher homotopy</a></p>
5341 </li>
5342
5343 <li>
5344 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p>
5345 </li>
5346
5347 <li>
5348 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/spherical+object'>spherical object and Pi(A)-algebra</a></p>
5349 </li>
5350
5351 <li>
5352 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p>
5353
5354 <ul>
5355 <li>
5356 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></p>
5357
5358 <ul>
5359 <li>
5360 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p>
5361 </li>
5362
5363 <li>
5364 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/cofibration+category'>cofibration category</a></p>
5365 </li>
5366
5367 <li>
5368 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Waldhausen category</a></p>
5369 </li>
5370 </ul>
5371 </li>
5372
5373 <li>
5374 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></p>
5375
5376 <ul>
5377 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a></li>
5378 </ul>
5379 </li>
5380 </ul>
5381 </li>
5382
5383 <li>
5384 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
5385
5386 <ul>
5387 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li>
5388 </ul>
5389 </li>
5390 </ul>
5391
5392 <p><strong>Paths and cylinders</strong></p>
5393
5394 <ul>
5395 <li>
5396 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a></p>
5397
5398 <ul>
5399 <li>
5400 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a></p>
5401 </li>
5402
5403 <li>
5404 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a></p>
5405 </li>
5406 </ul>
5407 </li>
5408
5409 <li>
5410 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
5411
5412 <ul>
5413 <li>
5414 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path object</a></p>
5415 </li>
5416
5417 <li>
5418 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
5419 </li>
5420
5421 <li>
5422 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle'>universal bundle</a></p>
5423 </li>
5424 </ul>
5425 </li>
5426
5427 <li>
5428 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a></p>
5429
5430 <ul>
5431 <li>
5432 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometric+homotopies'>homotopy localization</a></p>
5433 </li>
5434
5435 <li>
5436 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+object'>infinitesimal interval object</a></p>
5437 </li>
5438 </ul>
5439 </li>
5440 </ul>
5441
5442 <p><strong>Homotopy groups</strong></p>
5443
5444 <ul>
5445 <li>
5446 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
5447
5448 <ul>
5449 <li>
5450 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a></p>
5451
5452 <ul>
5453 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+topos'>fundamental group of a topos</a></li>
5454 </ul>
5455 </li>
5456
5457 <li>
5458 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p>
5459 </li>
5460
5461 <li>
5462 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p>
5463 </li>
5464
5465 <li>
5466 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p>
5467 </li>
5468 </ul>
5469 </li>
5470
5471 <li>
5472 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p>
5473
5474 <ul>
5475 <li>
5476 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a></p>
5477
5478 <ul>
5479 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path groupoid</a></li>
5480 </ul>
5481 </li>
5482
5483 <li>
5484 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p>
5485 </li>
5486
5487 <li>
5488 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p>
5489 </li>
5490 </ul>
5491 </li>
5492
5493 <li>
5494 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p>
5495
5496 <ul>
5497 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental category</a></li>
5498 </ul>
5499 </li>
5500 </ul>
5501
5502 <p><strong>Basic facts</strong></p>
5503
5504 <ul>
5505 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li>
5506 </ul>
5507
5508 <p><strong>Theorems</strong></p>
5509
5510 <ul>
5511 <li>
5512 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
5513 </li>
5514
5515 <li>
5516 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
5517 </li>
5518
5519 <li>
5520 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p>
5521 </li>
5522
5523 <li>
5524 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p>
5525 </li>
5526
5527 <li>
5528 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
5529 </li>
5530
5531 <li>
5532 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead&#39;s theorem</a></p>
5533 </li>
5534
5535 <li>
5536 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewicz theorem</a></p>
5537 </li>
5538
5539 <li>
5540 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois theory</a></p>
5541 </li>
5542
5543 <li>
5544 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p>
5545 </li>
5546 </ul>
5547 </div>
5548
5549 <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-Topos Theory</h4>
5550
5551 <div class='hide'>
5552 <p><strong><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos+theory'>(∞,1)-topos theory</a></strong></p>
5553
5554 <h2 id='background'>Background</h2>
5555
5556 <ul>
5557 <li>
5558 <p><a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>sheaf and topos theory</a></p>
5559 </li>
5560
5561 <li>
5562 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
5563 </li>
5564
5565 <li>
5566 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p>
5567 </li>
5568
5569 <li>
5570 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a></p>
5571 </li>
5572
5573 <li>
5574 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p>
5575 </li>
5576 </ul>
5577
5578 <h2 id='definitions'>Definitions</h2>
5579
5580 <ul>
5581 <li>
5582 <p><a class='existingWikiWord' href='/nlab/show/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-topos</a></p>
5583 </li>
5584
5585 <li>
5586 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'>(∞,1)-site</a></p>
5587 </li>
5588
5589 <li>
5590 <p><a class='existingWikiWord' href='/nlab/show/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p>
5591
5592 <ul>
5593 <li>
5594 <p><a class='existingWikiWord' href='/nlab/show/localization+of+an+%28infinity%2C1%29-category'>localization of an (∞,1)-category</a></p>
5595 </li>
5596
5597 <li>
5598 <p><a class='existingWikiWord' href='/nlab/show/topological+localization'>topological localization</a></p>
5599 </li>
5600
5601 <li>
5602 <p><a class='existingWikiWord' href='/nlab/show/hypercompletion'>hypercompletion</a></p>
5603 </li>
5604 </ul>
5605 </li>
5606
5607 <li>
5608 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></p>
5609
5610 <ul>
5611 <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a>/<a class='existingWikiWord' href='/nlab/show/derived+stack'>derived stack</a></li>
5612 </ul>
5613 </li>
5614
5615 <li>
5616 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p>
5617 </li>
5618
5619 <li>
5620 <p><a class='existingWikiWord' href='/nlab/show/%28n%2C1%29-topos'>(n,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/n-topos'>n-topos</a></p>
5621
5622 <ul>
5623 <li>
5624 <p><a class='existingWikiWord' href='/nlab/show/truncated+object'>n-truncated object</a></p>
5625 </li>
5626
5627 <li>
5628 <p><a class='existingWikiWord' href='/nlab/show/connected+object'>n-connected object</a></p>
5629 </li>
5630
5631 <li>
5632 <p><a class='existingWikiWord' href='/nlab/show/topos'>(1,1)-topos</a></p>
5633
5634 <ul>
5635 <li>
5636 <p><a class='existingWikiWord' href='/nlab/show/presheaf'>presheaf</a></p>
5637 </li>
5638
5639 <li>
5640 <p><a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a></p>
5641 </li>
5642 </ul>
5643 </li>
5644
5645 <li>
5646 <p><a class='existingWikiWord' href='/nlab/show/2-topos'>(2,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a></p>
5647
5648 <ul>
5649 <li><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-presheaf'>(2,1)-presheaf</a></li>
5650 </ul>
5651 </li>
5652 </ul>
5653 </li>
5654
5655 <li>
5656 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-quasitopos'>(∞,1)-quasitopos</a></p>
5657
5658 <ul>
5659 <li>
5660 <p><a class='existingWikiWord' href='/nlab/show/separated+%28infinity%2C1%29-presheaf'>separated (∞,1)-presheaf</a></p>
5661 </li>
5662
5663 <li>
5664 <p><a class='existingWikiWord' href='/nlab/show/quasitopos'>quasitopos</a></p>
5665
5666 <ul>
5667 <li><a class='existingWikiWord' href='/nlab/show/separated+presheaf'>separated presheaf</a></li>
5668 </ul>
5669 </li>
5670
5671 <li>
5672 <p><span class='newWikiWord'>(2,1)-quasitopos<a href='/nlab/new/%282%2C1%29-quasitopos'>?</a></span></p>
5673
5674 <ul>
5675 <li><a class='existingWikiWord' href='/nlab/show/separated+%282%2C1%29-presheaf'>separated (2,1)-presheaf</a></li>
5676 </ul>
5677 </li>
5678 </ul>
5679 </li>
5680
5681 <li>
5682 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-topos'>(∞,2)-topos</a></p>
5683 </li>
5684
5685 <li>
5686 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-topos'>(∞,n)-topos</a></p>
5687 </li>
5688 </ul>
5689
5690 <h2 id='characterization'>Characterization</h2>
5691
5692 <ul>
5693 <li>
5694 <p><a class='existingWikiWord' href='/nlab/show/pullback-stable+colimit'>universal colimits</a></p>
5695 </li>
5696
5697 <li>
5698 <p><a class='existingWikiWord' href='/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p>
5699 </li>
5700
5701 <li>
5702 <p><a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-topos</a></p>
5703
5704 <ul>
5705 <li><a class='existingWikiWord' href='/nlab/show/effective+epimorphism'>effective epimorphism</a></li>
5706 </ul>
5707 </li>
5708 </ul>
5709
5710 <h2 id='morphisms'>Morphisms</h2>
5711
5712 <ul>
5713 <li>
5714 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a></p>
5715 </li>
5716
5717 <li>
5718 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29Topos'>(∞,1)Topos</a></p>
5719 </li>
5720
5721 <li>
5722 <p><a class='existingWikiWord' href='/nlab/show/Lawvere+distribution'>Lawvere distribution</a></p>
5723 </li>
5724 </ul>
5725
5726 <h2 id='extra_stuff_structure_and_property'>Extra stuff, structure and property</h2>
5727
5728 <ul>
5729 <li>
5730 <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a></p>
5731
5732 <ul>
5733 <li>
5734 <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+object'>hypercomplete object</a></p>
5735 </li>
5736
5737 <li>
5738 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead theorem</a></p>
5739 </li>
5740 </ul>
5741 </li>
5742
5743 <li>
5744 <p><a class='existingWikiWord' href='/nlab/show/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></p>
5745 </li>
5746
5747 <li>
5748 <p><a class='existingWikiWord' href='/nlab/show/n-localic+%28infinity%2C1%29-topos'>n-localic (∞,1)-topos</a></p>
5749 </li>
5750
5751 <li>
5752 <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>locally n-connected (n,1)-topos</a></p>
5753 </li>
5754
5755 <li>
5756 <p><a class='existingWikiWord' href='/nlab/show/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a></p>
5757
5758 <ul>
5759 <li><a class='existingWikiWord' href='/nlab/show/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>geometry (for structured (∞,1)-toposes)</a></li>
5760 </ul>
5761 </li>
5762
5763 <li>
5764 <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>locally ∞-connected (∞,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>∞-connected (∞,1)-topos</a></p>
5765 </li>
5766
5767 <li>
5768 <p><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-local+geometric+morphism'>local (∞,1)-topos</a></p>
5769
5770 <ul>
5771 <li><a class='existingWikiWord' href='/nlab/show/concrete+%28infinity%2C1%29-sheaf'>concrete (∞,1)-sheaf</a></li>
5772 </ul>
5773 </li>
5774
5775 <li>
5776 <p><a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></p>
5777 </li>
5778 </ul>
5779
5780 <h2 id='models'>Models</h2>
5781
5782 <ul>
5783 <li>
5784 <p><a class='existingWikiWord' href='/nlab/show/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a></p>
5785
5786 <ul>
5787 <li>
5788 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p>
5789 </li>
5790
5791 <li>
5792 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+functors'>model structure on functors</a></p>
5793 </li>
5794
5795 <li>
5796 <p><a class='existingWikiWord' href='/nlab/show/model+site'>model site</a>/<a class='existingWikiWord' href='/nlab/show/sSet-site'>sSet-site</a></p>
5797 </li>
5798
5799 <li>
5800 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a></p>
5801 </li>
5802
5803 <li>
5804 <p><a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a></p>
5805 </li>
5806
5807 <li>
5808 <p><a class='existingWikiWord' href='/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves'>descent for presheaves with values in strict ∞-groupoids</a></p>
5809 </li>
5810 </ul>
5811 </li>
5812 </ul>
5813
5814 <h2 id='constructions'>Constructions</h2>
5815
5816 <p><strong>structures in a <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></strong></p>
5817
5818 <ul>
5819 <li>
5820 <p><a class='existingWikiWord' href='/nlab/show/shape+of+an+%28infinity%2C1%29-topos'>shape</a> / <a class='existingWikiWord' href='/nlab/show/coshape+of+an+%28infinity%2C1%29-topos'>coshape</a></p>
5821 </li>
5822
5823 <li>
5824 <p><a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a></p>
5825 </li>
5826
5827 <li>
5828 <p><a class='existingWikiWord' href='/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy</a></p>
5829
5830 <ul>
5831 <li>
5832 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>of a locally ∞-connected (∞,1)-topos</a></p>
5833 </li>
5834
5835 <li>
5836 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical</a>/<a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric</a> homotopy groups</p>
5837 </li>
5838
5839 <li>
5840 <p><a class='existingWikiWord' href='/nlab/show/Postnikov+tower+in+an+%28infinity%2C1%29-category'>Postnikov tower</a></p>
5841 </li>
5842
5843 <li>
5844 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+tower+in+an+%28infinity%2C1%29-topos'>Whitehead tower</a></p>
5845 </li>
5846 </ul>
5847 </li>
5848
5849 <li>
5850 <p><a class='existingWikiWord' href='/nlab/show/function+algebras+on+infinity-stacks'>rational homotopy</a></p>
5851 </li>
5852
5853 <li>
5854 <p><a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a></p>
5855
5856 <ul>
5857 <li>
5858 <p><a class='existingWikiWord' href='/nlab/show/homotopy+dimension'>homotopy dimension</a></p>
5859 </li>
5860
5861 <li>
5862 <p><a class='existingWikiWord' href='/nlab/show/cohomological+dimension'>cohomological dimension</a></p>
5863 </li>
5864
5865 <li>
5866 <p><a class='existingWikiWord' href='/nlab/show/covering+dimension'>covering dimension</a></p>
5867 </li>
5868
5869 <li>
5870 <p><a class='existingWikiWord' href='/nlab/show/Heyting+dimension'>Heyting dimension</a></p>
5871 </li>
5872 </ul>
5873 </li>
5874 </ul>
5875 <div>
5876 <p>
5877 <a href='/nlab/edit/%28infinity%2C1%29-topos+-+contents'>Edit this sidebar</a>
5878 </p>
5879 </div></div>
5880 </div>
5881 </div>
5882
5883 <h1 id='contents'>Contents</h1>
5884 <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#interpretation_as_stacks'>Interpretation as <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks</a></li><li><a href='#examples'>Examples</a></li><li><a href='#remarks'>Remarks</a></li><li><a href='#properties'>Properties</a></li><li><a href='#related_entries'>Related entries</a></li><li><a href='#references'>References</a></li></ul></div>
5885 <h2 id='definition'>Definition</h2>
5886
5887 <p><em>Simplicial presheaves</em> over some <a class='existingWikiWord' href='/nlab/show/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> are</p>
5888
5889 <ul>
5890 <li><a class='existingWikiWord' href='/nlab/show/presheaf'>Presheaves</a> with values in the category <a class='existingWikiWord' href='/nlab/show/SimpSet'>SimpSet</a> of simplicial sets, i.e., functors <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>S^{op} \to \Simp\Set</annotation></semantics></math>, i.e., functors <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>S^{op} \to [\Delta^{op}, \Set]</annotation></semantics></math>;</li>
5891 </ul>
5892
5893 <p>or equivalently, using the Hom-<a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjunction</a> and symmetry of the <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed monoidal structure</a> on <a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a></p>
5894
5895 <ul>
5896 <li>simplicial objects in the category of presheaves, i.e. functors <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^{op} \to [S^{op},\Set]</annotation></semantics></math>.</li>
5897 </ul>
5898
5899 <h2 id='interpretation_as_stacks'>Interpretation as <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks</h2>
5900
5901 <p>Regarding <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>\Simp\Set</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> using the standard <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+sets'>model structure on simplicial sets</a> and inducing from that a model structure on <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S^{op}, \Simp\Set]</annotation></semantics></math> makes simplicial presheaves a model for <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/infinity-stack'>stacks</a>, as described at <a class='existingWikiWord' href='/nlab/show/infinity-stack+homotopically'>infinity-stack homotopically</a>.</p>
5902
5903 <p>In more illustrative language this means that a simplicial presheaf on <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> can be regarded as an <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/infinity-groupoid'>groupoid</a> (in particular a <a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complex</a>) whose space of <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-morphisms is modeled on the objects of <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> in the sense described at <a class='existingWikiWord' href='/nlab/show/space+and+quantity'>space and quantity</a>.</p>
5904
5905 <h2 id='examples'>Examples</h2>
5906
5907 <ul>
5908 <li>
5909 <p>Notice that most definitions of <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/infinity-category'>category</a> the <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-category is itself defined to be a <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial set</a> with extra structure (in a <a class='existingWikiWord' href='/nlab/show/geometric+definition+of+higher+categories'>geometric definition of higher category</a>) or gives rise to a simplicial set under taking its <a class='existingWikiWord' href='/nlab/show/nerve'>nerve</a> (in an <a class='existingWikiWord' href='/nlab/show/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a>). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.</p>
5910 </li>
5911
5912 <li>
5913 <p>In particular, regarding a <a class='existingWikiWord' href='/nlab/show/group'>group</a> <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> as a one object category <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}G</annotation></semantics></math> and then taking the nerve <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>N(\mathbf{B}G) \in \Simp\Set</annotation></semantics></math> of these (the “classifying simplicial set of the group whose <a class='existingWikiWord' href='/nlab/show/geometric+realization'>geometric realization</a> is the <a class='existingWikiWord' href='/nlab/show/classifying+space'>classifying space</a> <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}G</annotation></semantics></math>), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.</p>
5914 </li>
5915 </ul>
5916
5917 <h2 id='remarks'>Remarks</h2>
5918
5919 <ul>
5920 <li>There are various useful <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> structures on the category of simplicial presheaves. See <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a>.</li>
5921 </ul>
5922
5923 <h2 id='properties'>Properties</h2>
5924
5925 <p>Here are some basic but useful facts about simplicial presheaves.</p>
5926
5927 <div class='un_prop'>
5928 <h6 id='proposition'>Proposition</h6>
5929
5930 <p>Every simplicial presheaf <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy colimit</a> over a <a class='existingWikiWord' href='/nlab/show/diagram'>diagram</a> of <a class='existingWikiWord' href='/nlab/show/Set'>Set</a>-valued sheaves regarded as discrete simplicial sheaves.</p>
5931
5932 <p>More precisely, for <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>X : S^{op} \to SSet</annotation></semantics></math> a simplicial presheaf, let <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>↪</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mi>SSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>D_X : \Delta^{op} \to [S^{op},Set] \hookrightarrow [S^{op},SSet]</annotation></semantics></math> be given by <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>↦</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>D_X : [n] \mapsto X_n</annotation></semantics></math>. Then there is a weak equivalence</p>
5933 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hocolim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
5934 hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X
5935 \,.
5936
5937 </annotation></semantics></math></div></div>
5938
5939 <div class='proof'>
5940 <h6 id='proof'>Proof</h6>
5941
5942 <p>See for instance <a href='http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6'>remark 2.1, p. 6</a></p>
5943
5944 <ul>
5945 <li><a class='existingWikiWord' href='/nlab/show/Daniel+Dugger'>Daniel Dugger</a>, <a class='existingWikiWord' href='/nlab/show/Sharon+Hollander'>Sharon Hollander</a>, <a class='existingWikiWord' href='/nlab/show/Daniel+Isaksen'>Daniel Isaksen</a>, <em>Hypercovers and simplicial presheaves</em>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 136 Issue 1, 2004 (<a href='https://arxiv.org/abs/math/0205027'>arXiv:math/0205027</a>, <a href='http://www.math.uiuc.edu/K-theory/0563'>K-theory:0563</a>, <a href='https://doi.org/10.1017/S0305004103007175'>doi:10.1017/S0305004103007175</a>)</li>
5946 </ul>
5947
5948 <p>(which is otherwise about <a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>).</p>
5949 </div>
5950
5951 <div class='un_cor'>
5952 <h6 id='corollary'>Corollary</h6>
5953
5954 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>:</mo><mo stretchy='false'>(</mo><msup><mi>SSet</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>×</mo><msup><mi>SSet</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>[-,-] : (SSet^{S^{op}})^{op} \times SSet^{S^{op}} \to SSet</annotation></semantics></math> be the canonical <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>SSet</annotation></semantics></math>-enrichment of the category of simplicial presheaves (i.e. the assignment of <a class='existingWikiWord' href='/nlab/show/SimpSet'>SSet</a>-<a class='existingWikiWord' href='/nlab/show/enriched+functor+category'>enriched functor categories</a>).</p>
5955
5956 <p>It follows in particular from the above that every such <a class='existingWikiWord' href='/nlab/show/hom-object'>hom-object</a> <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,A]</annotation></semantics></math> of simplical presheaves can be written as a <a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy limit</a> (in <a class='existingWikiWord' href='/nlab/show/SimpSet'>SSet</a> for instance realized as a <a class='existingWikiWord' href='/nlab/show/weighted+limit'>weighted limit</a>, as described there) over evaluations of <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p>
5957 </div>
5958
5959 <div class='proof'>
5960 <h6 id='proof_2'>Proof</h6>
5961
5962 <p>First the above yields</p>
5963 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd> <mtd><mo>≃</mo><mo stretchy='false'>[</mo><msub><mi>hocolim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy='false'>[</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
5964 \begin{aligned}
5965 [X, A ] &amp; \simeq [ hocolim_{[n] \in \Delta} X_n , A ]
5966 \\
5967 &amp; holim_{[n] \in \Delta} [X_n, A]
5968 \end{aligned}
5969 \,.
5970
5971 </annotation></semantics></math></div>
5972 <p>Next from the <a class='existingWikiWord' href='/nlab/show/co-Yoneda+lemma'>co-Yoneda lemma</a> we know that the <a class='existingWikiWord' href='/nlab/show/Set'>Set</a>-valued presheaves <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>X_n</annotation></semantics></math> are in turn colimits over representables in <math class='maruku-mathml' display='inline' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, so that</p>
5973 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>≃</mo><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy='false'>[</mo><msub><mi>colim</mi> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>lim</mi> <mi>i</mi></msub><mo stretchy='false'>[</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
5974 \begin{aligned}
5975 \cdots &amp; \simeq
5976 holim_{[n] \in \Delta}
5977 [ colim_i U_{i}, A]
5978 \\
5979 &amp; \simeq
5980 holim_{[n] \in \Delta} lim_i
5981 [ U_{i}, A]
5982 \end{aligned}
5983 \,.
5984
5985 </annotation></semantics></math></div>
5986 <p>And finally the <a class='existingWikiWord' href='/nlab/show/Yoneda+lemma'>Yoneda lemma</a> reduces this to</p>
5987 <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ebe0afcebcbc829252dcf548033d67aad4bfd643_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>⋯</mi></mtd> <mtd><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>lim</mi> <mi>i</mi></msub><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'>
5988 \begin{aligned}
5989 \cdots
5990 &amp;
5991 holim_{[n] \in \Delta} lim_i
5992 A(U_i)
5993 \end{aligned}
5994 \,.
5995
5996 </annotation></semantics></math></div></div>
5997
5998 <p>Notice that these kinds of computations are in particular often used when checking/computing <a class='existingWikiWord' href='/nlab/show/descent'>descent and codescent</a> along a <a class='existingWikiWord' href='/nlab/show/cover'>cover</a> or <a class='existingWikiWord' href='/nlab/show/hypercover'>hypercover</a>. For more on that in the context of simplicial presheaves see <a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>.</p>
5999
6000 <h2 id='related_entries'>Related entries</h2>
6001
6002 <ul>
6003 <li>
6004 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a></p>
6005 </li>
6006
6007 <li>
6008 <p><a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a></p>
6009 </li>
6010
6011 <li>
6012 <p><a class='existingWikiWord' href='/nlab/show/sheaf+of+spectra'>presheaf of spectra</a></p>
6013 </li>
6014 </ul>
6015
6016 <p>Applications appear for instance at</p>
6017
6018 <ul>
6019 <li><a class='existingWikiWord' href='/nlab/show/geometric+infinity-function+theory'>geometric infinity-function theory</a></li>
6020 </ul>
6021
6022 <h2 id='references'>References</h2>
6023
6024 <p>The original articles are</p>
6025
6026 <ul>
6027 <li>
6028 <p><a class='existingWikiWord' href='/nlab/show/Kenneth+Brown'>Kenneth S. Brown</a>, <em>Abstract homotopy theory and generalized sheaf cohomology</em>. Transactions of the American Mathematical Society 186 (1973), 419-419. <a href='http://dx.doi.org/10.1090/s0002-9947-1973-0341469-9'>doi</a>.</p>
6029 </li>
6030
6031 <li>
6032 <p><a class='existingWikiWord' href='/nlab/show/Kenneth+Brown'>Kenneth S. Brown</a>, <a class='existingWikiWord' href='/nlab/show/Stephen+M.+Gersten'>Stephen M. Gersten</a>, <em>Algebraic K-theory as generalized sheaf cohomology</em>. In: Higher K-Theories. Lecture Notes in Mathematics (1973), 266–292. <a href='http://dx.doi.org/10.1007/bfb0067062'>doi</a>.</p>
6033 </li>
6034
6035 <li>
6036 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Simplicial objects in a Grothendieck topos</em>. In: Applications of algebraic K-theory to algebraic geometry and number theory. Contemporary Mathematics (1986), 193-239. <a href='http://dx.doi.org/10.1090/conm/055.1/862637'>doi</a></p>
6037 </li>
6038
6039 <li>
6040 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Simplical presheaves</em>. Journal of Pure and Applied Algebra 47:1 (1987), 35-87. <a href='http://dx.doi.org/10.1016/0022-4049(87)90100-9'>doi</a></p>
6041 </li>
6042 </ul>
6043
6044 <p>A modern expository account is</p>
6045
6046 <ul>
6047 <li><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>John F. Jardine</a>, <em>Local Homotopy Theory</em>, Springer, 2015. <a href='http://dx.doi.org/10.1007/978-1-4939-2300-7'>doi</a>.</li>
6048 </ul>
6049
6050 <p>Further articles:</p>
6051
6052 <ul>
6053 <li>
6054 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Stacks and the homotopy theory of simplicial sheaves</em>. Homology, Homotopy and Applications 3:2 (2001), 361-384. <a href='http://dx.doi.org/10.4310/hha.2001.v3.n2.a5'>doi</a>.</p>
6055 </li>
6056
6057 <li>
6058 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Fields Lectures: Simplicial presheaves</em>. <a href='https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf'>PDF</a>.</p>
6059 </li>
6060 </ul>
6061
6062 <p>For their interpretation in the more general context of <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(infinity,1)-sheaves</a> see Section 6.5.2 of</p>
6063
6064 <ul>
6065 <li><a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/Higher+Topos+Theory'>Higher Topos Theory</a>.</li>
6066 </ul>
6067
6068 <p>
6069 </p>
6070
6071 <p>
6072
6073 </p>
6074
6075 <p>
6076 </p> </div>
6077 </content>
6078 </entry>
6079 <entry>
6080 <title type="html">Milky Way</title>
6081 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Milky+Way"/>
6082 <updated>2021-07-01T10:22:24Z</updated>
6083 <published>2019-04-10T13:55:53Z</published>
6084 <id>tag:ncatlab.org,2019-04-10:nLab,Milky+Way</id>
6085 <author>
6086 <name>Urs Schreiber</name>
6087 </author>
6088 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Milky+Way">
6089 <div xmlns="http://www.w3.org/1999/xhtml">
6090 <div class='rightHandSide'>
6091 <div class='toc clickDown' tabindex='0'>
6092 <h3 id='context'>Context</h3>
6093
6094 <h4 id='physics'>Physics</h4>
6095
6096 <div class='hide'>
6097 <p><strong><a class='existingWikiWord' href='/nlab/show/physics'>physics</a></strong>, <a class='existingWikiWord' href='/nlab/show/mathematical+physics'>mathematical physics</a>, <a class='existingWikiWord' href='/nlab/show/philosophy+of+physics'>philosophy of physics</a></p>
6098
6099 <h2 id='surveys_textbooks_and_lecture_notes'>Surveys, textbooks and lecture notes</h2>
6100
6101 <ul>
6102 <li>
6103 <p><em><a class='existingWikiWord' href='/nlab/show/higher+category+theory+and+physics'>(higher) category theory and physics</a></em></p>
6104 </li>
6105
6106 <li>
6107 <p><em><a class='existingWikiWord' href='/nlab/show/geometry+of+physics'>geometry of physics</a></em></p>
6108 </li>
6109
6110 <li>
6111 <p><a class='existingWikiWord' href='/nlab/show/books+and+reviews+in+mathematical+physics'>books and reviews</a>, <a class='existingWikiWord' href='/nlab/show/physics+resources'>physics resources</a></p>
6112 </li>
6113 </ul>
6114 <hr/>
6115 <p><a class='existingWikiWord' href='/nlab/show/theory+%28physics%29'>theory (physics)</a>, <a class='existingWikiWord' href='/nlab/show/model+%28in+theoretical+physics%29'>model (physics)</a></p>
6116
6117 <p><a class='existingWikiWord' href='/nlab/show/experimental+observation'>experiment</a>, <a class='existingWikiWord' href='/nlab/show/measurement'>measurement</a>, <a class='existingWikiWord' href='/nlab/show/computable+physics'>computable physics</a></p>
6118
6119 <ul>
6120 <li>
6121 <p><strong><a class='existingWikiWord' href='/nlab/show/mechanics'>mechanics</a></strong></p>
6122
6123 <ul>
6124 <li>
6125 <p><a class='existingWikiWord' href='/nlab/show/mass'>mass</a>, <a class='existingWikiWord' href='/nlab/show/charge'>charge</a>, <a class='existingWikiWord' href='/nlab/show/momentum'>momentum</a>, <a class='existingWikiWord' href='/nlab/show/angular+momentum'>angular momentum</a>, <a class='existingWikiWord' href='/nlab/show/moment+of+inertia'>moment of inertia</a></p>
6126 </li>
6127
6128 <li>
6129 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian+dynamics+on+Lie+groups'>dynamics on Lie groups</a></p>
6130
6131 <ul>
6132 <li><a class='existingWikiWord' href='/nlab/show/rigid+body+dynamics'>rigid body dynamics</a></li>
6133 </ul>
6134 </li>
6135 </ul>
6136 </li>
6137
6138 <li>
6139 <p><a class='existingWikiWord' href='/nlab/show/field+%28physics%29'>field (physics)</a></p>
6140
6141 <ul>
6142 <li>
6143 <p><a class='existingWikiWord' href='/nlab/show/Lagrangian+density'>Lagrangian mechanics</a></p>
6144
6145 <ul>
6146 <li>
6147 <p><a class='existingWikiWord' href='/nlab/show/configuration+space'>configuration space</a>, <a class='existingWikiWord' href='/nlab/show/state'>state</a></p>
6148 </li>
6149
6150 <li>
6151 <p><a class='existingWikiWord' href='/nlab/show/action+functional'>action functional</a>, <a class='existingWikiWord' href='/nlab/show/Lagrangian+density'>Lagrangian</a></p>
6152 </li>
6153
6154 <li>
6155 <p><a class='existingWikiWord' href='/nlab/show/phase+space'>covariant phase space</a>, <a class='existingWikiWord' href='/nlab/show/Euler-Lagrange+equation'>Euler-Lagrange equations</a></p>
6156 </li>
6157 </ul>
6158 </li>
6159
6160 <li>
6161 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian+mechanics'>Hamiltonian mechanics</a></p>
6162
6163 <ul>
6164 <li>
6165 <p><a class='existingWikiWord' href='/nlab/show/phase+space'>phase space</a></p>
6166 </li>
6167
6168 <li>
6169 <p><a class='existingWikiWord' href='/nlab/show/symplectic+geometry'>symplectic geometry</a></p>
6170
6171 <ul>
6172 <li>
6173 <p><a class='existingWikiWord' href='/nlab/show/Poisson+manifold'>Poisson manifold</a></p>
6174 </li>
6175
6176 <li>
6177 <p><a class='existingWikiWord' href='/nlab/show/symplectic+manifold'>symplectic manifold</a></p>
6178 </li>
6179
6180 <li>
6181 <p><a class='existingWikiWord' href='/nlab/show/symplectic+groupoid'>symplectic groupoid</a></p>
6182 </li>
6183 </ul>
6184 </li>
6185
6186 <li>
6187 <p><a class='existingWikiWord' href='/nlab/show/multisymplectic+geometry'>multisymplectic geometry</a></p>
6188
6189 <ul>
6190 <li><a class='existingWikiWord' href='/nlab/show/symplectic+Lie+n-algebroid'>n-symplectic manifold</a></li>
6191 </ul>
6192 </li>
6193 </ul>
6194 </li>
6195
6196 <li>
6197 <p><a class='existingWikiWord' href='/nlab/show/spacetime'>spacetime</a></p>
6198
6199 <ul>
6200 <li>
6201 <p><a class='existingWikiWord' href='/nlab/show/smooth+Lorentzian+space'>smooth Lorentzian manifold</a></p>
6202 </li>
6203
6204 <li>
6205 <p><a class='existingWikiWord' href='/nlab/show/special+relativity'>special relativity</a></p>
6206 </li>
6207
6208 <li>
6209 <p><a class='existingWikiWord' href='/nlab/show/general+relativity'>general relativity</a></p>
6210 </li>
6211
6212 <li>
6213 <p><a class='existingWikiWord' href='/nlab/show/gravity'>gravity</a></p>
6214
6215 <ul>
6216 <li>
6217 <p><a class='existingWikiWord' href='/nlab/show/supergravity'>supergravity</a>, <a class='existingWikiWord' href='/nlab/show/dilaton'>dilaton gravity</a></p>
6218 </li>
6219
6220 <li>
6221 <p><a class='existingWikiWord' href='/nlab/show/black+hole'>black hole</a></p>
6222 </li>
6223 </ul>
6224 </li>
6225 </ul>
6226 </li>
6227 </ul>
6228 </li>
6229
6230 <li>
6231 <p><strong><a class='existingWikiWord' href='/nlab/show/classical+field+theory'>Classical field theory</a></strong></p>
6232
6233 <ul>
6234 <li>
6235 <p><a class='existingWikiWord' href='/nlab/show/classical+physics'>classical physics</a></p>
6236
6237 <ul>
6238 <li><a class='existingWikiWord' href='/nlab/show/classical+mechanics'>classical mechanics</a></li>
6239
6240 <li><a class='existingWikiWord' href='/nlab/show/wave'>waves</a> and <a class='existingWikiWord' href='/nlab/show/optics'>optics</a></li>
6241
6242 <li><a class='existingWikiWord' href='/nlab/show/thermodynamics'>thermodynamics</a></li>
6243 </ul>
6244 </li>
6245 </ul>
6246 </li>
6247
6248 <li>
6249 <p><strong><a class='existingWikiWord' href='/nlab/show/quantum+mechanics'>Quantum Mechanics</a></strong></p>
6250
6251 <ul>
6252 <li>
6253 <p><a class='existingWikiWord' href='/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories'>in terms of ∞-compact categories</a></p>
6254 </li>
6255
6256 <li>
6257 <p><a class='existingWikiWord' href='/nlab/show/quantum+information'>quantum information</a></p>
6258 </li>
6259
6260 <li>
6261 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian'>Hamiltonian operator</a></p>
6262 </li>
6263
6264 <li>
6265 <p><a class='existingWikiWord' href='/nlab/show/density+matrix'>density matrix</a></p>
6266 </li>
6267
6268 <li>
6269 <p><a class='existingWikiWord' href='/nlab/show/Kochen-Specker+theorem'>Kochen-Specker theorem</a></p>
6270 </li>
6271
6272 <li>
6273 <p><a class='existingWikiWord' href='/nlab/show/Bell%27s+theorem'>Bell&#39;s theorem</a></p>
6274 </li>
6275
6276 <li>
6277 <p><a class='existingWikiWord' href='/nlab/show/Gleason%27s+theorem'>Gleason&#39;s theorem</a></p>
6278 </li>
6279 </ul>
6280 </li>
6281
6282 <li>
6283 <p><strong><a class='existingWikiWord' href='/nlab/show/quantization'>Quantization</a></strong></p>
6284
6285 <ul>
6286 <li>
6287 <p><a class='existingWikiWord' href='/nlab/show/geometric+quantization'>geometric quantization</a></p>
6288 </li>
6289
6290 <li>
6291 <p><a class='existingWikiWord' href='/nlab/show/deformation+quantization'>deformation quantization</a></p>
6292 </li>
6293
6294 <li>
6295 <p><a class='existingWikiWord' href='/nlab/show/path+integral'>path integral quantization</a></p>
6296 </li>
6297
6298 <li>
6299 <p><a class='existingWikiWord' href='/nlab/show/semiclassical+approximation'>semiclassical approximation</a></p>
6300 </li>
6301 </ul>
6302 </li>
6303
6304 <li>
6305 <p><strong><a class='existingWikiWord' href='/nlab/show/quantum+field+theory'>Quantum Field Theory</a></strong></p>
6306
6307 <ul>
6308 <li>
6309 <p>Axiomatizations</p>
6310
6311 <ul>
6312 <li>
6313 <p><a class='existingWikiWord' href='/nlab/show/AQFT'>algebraic QFT</a></p>
6314
6315 <ul>
6316 <li>
6317 <p><a class='existingWikiWord' href='/nlab/show/Wightman+axioms'>Wightman axioms</a></p>
6318 </li>
6319
6320 <li>
6321 <p><a class='existingWikiWord' href='/nlab/show/Haag-Kastler+axioms'>Haag-Kastler axioms</a></p>
6322
6323 <ul>
6324 <li>
6325 <p><a class='existingWikiWord' href='/nlab/show/operator+algebra'>operator algebra</a></p>
6326 </li>
6327
6328 <li>
6329 <p><a class='existingWikiWord' href='/nlab/show/causally+local+net+of+observables'>local net</a></p>
6330 </li>
6331 </ul>
6332 </li>
6333
6334 <li>
6335 <p><a class='existingWikiWord' href='/nlab/show/conformal+net'>conformal net</a></p>
6336 </li>
6337
6338 <li>
6339 <p><a class='existingWikiWord' href='/nlab/show/Reeh-Schlieder+theorem'>Reeh-Schlieder theorem</a></p>
6340 </li>
6341
6342 <li>
6343 <p><a class='existingWikiWord' href='/nlab/show/Osterwalder-Schrader+theorem'>Osterwalder-Schrader theorem</a></p>
6344 </li>
6345
6346 <li>
6347 <p><a class='existingWikiWord' href='/nlab/show/PCT+theorem'>PCT theorem</a></p>
6348 </li>
6349
6350 <li>
6351 <p><a class='existingWikiWord' href='/nlab/show/Bisognano-Wichmann+theorem'>Bisognano-Wichmann theorem</a></p>
6352
6353 <ul>
6354 <li><a class='existingWikiWord' href='/nlab/show/modular+theory'>modular theory</a></li>
6355 </ul>
6356 </li>
6357
6358 <li>
6359 <p><a class='existingWikiWord' href='/nlab/show/spin-statistics+theorem'>spin-statistics theorem</a></p>
6360
6361 <ul>
6362 <li><a class='existingWikiWord' href='/nlab/show/boson'>boson</a>, <a class='existingWikiWord' href='/nlab/show/fermion'>fermion</a></li>
6363 </ul>
6364 </li>
6365 </ul>
6366 </li>
6367
6368 <li>
6369 <p><a class='existingWikiWord' href='/nlab/show/functorial+field+theory'>functorial QFT</a></p>
6370
6371 <ul>
6372 <li>
6373 <p><a class='existingWikiWord' href='/nlab/show/cobordism'>cobordism</a></p>
6374 </li>
6375
6376 <li>
6377 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a></p>
6378 </li>
6379
6380 <li>
6381 <p><a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>cobordism hypothesis</a>-theorem</p>
6382 </li>
6383
6384 <li>
6385 <p><a class='existingWikiWord' href='/nlab/show/extended+topological+quantum+field+theory'>extended topological quantum field theory</a></p>
6386 </li>
6387 </ul>
6388 </li>
6389 </ul>
6390 </li>
6391
6392 <li>
6393 <p>Tools</p>
6394
6395 <ul>
6396 <li>
6397 <p><a class='existingWikiWord' href='/nlab/show/perturbative+quantum+field+theory'>perturbative quantum field theory</a>, <a class='existingWikiWord' href='/nlab/show/vacuum'>vacuum</a></p>
6398 </li>
6399
6400 <li>
6401 <p><a class='existingWikiWord' href='/nlab/show/effective+quantum+field+theory'>effective quantum field theory</a></p>
6402 </li>
6403
6404 <li>
6405 <p><a class='existingWikiWord' href='/nlab/show/renormalization'>renormalization</a></p>
6406 </li>
6407
6408 <li>
6409 <p><a class='existingWikiWord' href='/nlab/show/BV-BRST+formalism'>BV-BRST formalism</a></p>
6410 </li>
6411
6412 <li>
6413 <p><a class='existingWikiWord' href='/nlab/show/geometric+infinity-function+theory'>geometric ∞-function theory</a></p>
6414 </li>
6415 </ul>
6416 </li>
6417
6418 <li>
6419 <p><a class='existingWikiWord' href='/nlab/show/particle+physics'>particle physics</a></p>
6420
6421 <ul>
6422 <li>
6423 <p><a class='existingWikiWord' href='/nlab/show/phenomenology'>phenomenology</a></p>
6424 </li>
6425
6426 <li>
6427 <p><a class='existingWikiWord' href='/nlab/show/model+%28in+theoretical+physics%29'>models</a></p>
6428
6429 <ul>
6430 <li>
6431 <p><a class='existingWikiWord' href='/nlab/show/standard+model+of+particle+physics'>standard model of particle physics</a></p>
6432 </li>
6433
6434 <li>
6435 <p><a class='existingWikiWord' href='/nlab/show/fields+and+quanta+-+table'>fields and quanta</a></p>
6436 </li>
6437
6438 <li>
6439 <p><a class='existingWikiWord' href='/nlab/show/GUT'>Grand Unified Theories</a>, <a class='existingWikiWord' href='/nlab/show/MSSM'>MSSM</a></p>
6440 </li>
6441 </ul>
6442 </li>
6443
6444 <li>
6445 <p><a class='existingWikiWord' href='/nlab/show/scattering+amplitude'>scattering amplitude</a></p>
6446
6447 <ul>
6448 <li><a class='existingWikiWord' href='/nlab/show/on-shell+recursion'>on-shell recursion</a>, <a class='existingWikiWord' href='/nlab/show/KLT+relations'>KLT relations</a></li>
6449 </ul>
6450 </li>
6451 </ul>
6452 </li>
6453
6454 <li>
6455 <p>Structural phenomena</p>
6456
6457 <ul>
6458 <li>
6459 <p><a class='existingWikiWord' href='/nlab/show/universality+class'>universality class</a></p>
6460 </li>
6461
6462 <li>
6463 <p><a class='existingWikiWord' href='/nlab/show/quantum+anomaly'>quantum anomaly</a></p>
6464
6465 <ul>
6466 <li><a class='existingWikiWord' href='/nlab/show/Green-Schwarz+mechanism'>Green-Schwarz mechanism</a></li>
6467 </ul>
6468 </li>
6469
6470 <li>
6471 <p><a class='existingWikiWord' href='/nlab/show/instanton'>instanton</a></p>
6472 </li>
6473
6474 <li>
6475 <p><a class='existingWikiWord' href='/nlab/show/spontaneously+broken+symmetry'>spontaneously broken symmetry</a></p>
6476 </li>
6477
6478 <li>
6479 <p><a class='existingWikiWord' href='/nlab/show/Kaluza-Klein+mechanism'>Kaluza-Klein mechanism</a></p>
6480 </li>
6481
6482 <li>
6483 <p><a class='existingWikiWord' href='/nlab/show/integrable+system'>integrable systems</a></p>
6484 </li>
6485
6486 <li>
6487 <p><a class='existingWikiWord' href='/nlab/show/holonomic+quantum+field'>holonomic quantum fields</a></p>
6488 </li>
6489 </ul>
6490 </li>
6491
6492 <li>
6493 <p>Types of quantum field thories</p>
6494
6495 <ul>
6496 <li>
6497 <p><a class='existingWikiWord' href='/nlab/show/topological+quantum+field+theory'>TQFT</a></p>
6498
6499 <ul>
6500 <li>
6501 <p><a class='existingWikiWord' href='/nlab/show/2d+TQFT'>2d TQFT</a></p>
6502 </li>
6503
6504 <li>
6505 <p><a class='existingWikiWord' href='/nlab/show/Dijkgraaf-Witten+theory'>Dijkgraaf-Witten theory</a></p>
6506 </li>
6507
6508 <li>
6509 <p><a class='existingWikiWord' href='/nlab/show/Chern-Simons+theory'>Chern-Simons theory</a></p>
6510 </li>
6511 </ul>
6512 </li>
6513
6514 <li>
6515 <p><a class='existingWikiWord' href='/nlab/show/TCFT'>TCFT</a></p>
6516
6517 <ul>
6518 <li>
6519 <p><a class='existingWikiWord' href='/nlab/show/A-model'>A-model</a>, <a class='existingWikiWord' href='/nlab/show/B-model'>B-model</a></p>
6520 </li>
6521
6522 <li>
6523 <p><a class='existingWikiWord' href='/nlab/show/mirror+symmetry'>homological mirror symmetry</a></p>
6524 </li>
6525 </ul>
6526 </li>
6527
6528 <li>
6529 <p><a class='existingWikiWord' href='/nlab/show/QFT+with+defects'>QFT with defects</a></p>
6530 </li>
6531
6532 <li>
6533 <p><a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>conformal field theory</a></p>
6534 </li>
6535
6536 <li>
6537 <p><a class='existingWikiWord' href='/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory'>(1,1)-dimensional Euclidean field theories and K-theory</a></p>
6538 </li>
6539
6540 <li>
6541 <p><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory'>(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p>
6542 </li>
6543
6544 <li>
6545 <p><a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>CFT</a></p>
6546
6547 <ul>
6548 <li>
6549 <p><a class='existingWikiWord' href='/nlab/show/Wess-Zumino-Witten+model'>WZW model</a></p>
6550 </li>
6551
6552 <li>
6553 <p><a class='existingWikiWord' href='/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT'>6d (2,0)-supersymmetric QFT</a></p>
6554 </li>
6555 </ul>
6556 </li>
6557
6558 <li>
6559 <p><a class='existingWikiWord' href='/nlab/show/gauge+theory'>gauge theory</a></p>
6560
6561 <ul>
6562 <li>
6563 <p><a class='existingWikiWord' href='/nlab/show/field+strength'>field strength</a></p>
6564 </li>
6565
6566 <li>
6567 <p><a class='existingWikiWord' href='/nlab/show/gauge+group'>gauge group</a>, <a class='existingWikiWord' href='/nlab/show/gauge+transformation'>gauge transformation</a>, <a class='existingWikiWord' href='/nlab/show/gauge+fixing'>gauge fixing</a></p>
6568 </li>
6569
6570 <li>
6571 <p>examples</p>
6572
6573 <ul>
6574 <li><a class='existingWikiWord' href='/nlab/show/electromagnetic+field'>electromagnetic field</a>, <a class='existingWikiWord' href='/nlab/show/quantum+electrodynamics'>QED</a></li>
6575 </ul>
6576 </li>
6577
6578 <li>
6579 <p><a class='existingWikiWord' href='/nlab/show/electric+charge'>electric charge</a></p>
6580 </li>
6581
6582 <li>
6583 <p><a class='existingWikiWord' href='/nlab/show/magnetic+charge'>magnetic charge</a></p>
6584
6585 <ul>
6586 <li><a class='existingWikiWord' href='/nlab/show/Yang-Mills+field'>Yang-Mills field</a>, <a class='existingWikiWord' href='/nlab/show/QCD'>QCD</a></li>
6587 </ul>
6588 </li>
6589
6590 <li>
6591 <p><a class='existingWikiWord' href='/nlab/show/Yang-Mills+theory'>Yang-Mills theory</a></p>
6592 </li>
6593
6594 <li>
6595 <p><a class='existingWikiWord' href='/nlab/show/The+Dirac+Electron'>spinors in Yang-Mills theory</a></p>
6596 </li>
6597
6598 <li>
6599 <p><a class='existingWikiWord' href='/nlab/show/topological+Yang-Mills+theory'>topological Yang-Mills theory</a></p>
6600
6601 <ul>
6602 <li><a class='existingWikiWord' href='/nlab/show/Kalb-Ramond+field'>Kalb-Ramond field</a></li>
6603
6604 <li><a class='existingWikiWord' href='/nlab/show/supergravity+C-field'>supergravity C-field</a></li>
6605
6606 <li><a class='existingWikiWord' href='/nlab/show/RR+field'>RR field</a></li>
6607
6608 <li><a class='existingWikiWord' href='/nlab/show/first-order+formulation+of+gravity'>first-order formulation of gravity</a></li>
6609 </ul>
6610 </li>
6611
6612 <li>
6613 <p><a class='existingWikiWord' href='/nlab/show/general+covariance'>general covariance</a></p>
6614 </li>
6615
6616 <li>
6617 <p><a class='existingWikiWord' href='/nlab/show/supergravity'>supergravity</a></p>
6618 </li>
6619
6620 <li>
6621 <p><a class='existingWikiWord' href='/nlab/show/D%27Auria-Fre+formulation+of+supergravity'>D&#39;Auria-Fre formulation of supergravity</a></p>
6622 </li>
6623
6624 <li>
6625 <p><a class='existingWikiWord' href='/nlab/show/gravity+as+a+BF+theory'>gravity as a BF-theory</a></p>
6626 </li>
6627 </ul>
6628 </li>
6629
6630 <li>
6631 <p><a class='existingWikiWord' href='/nlab/show/sigma-model'>sigma-model</a></p>
6632
6633 <ul>
6634 <li>
6635 <p><a class='existingWikiWord' href='/nlab/show/particle'>particle</a>, <a class='existingWikiWord' href='/nlab/show/relativistic+particle'>relativistic particle</a>, <a class='existingWikiWord' href='/nlab/show/fundamental+particle'>fundamental particle</a>, <a class='existingWikiWord' href='/nlab/show/spinning+particle'>spinning particle</a>, <a class='existingWikiWord' href='/nlab/show/superparticle'>superparticle</a></p>
6636 </li>
6637
6638 <li>
6639 <p><a class='existingWikiWord' href='/nlab/show/string'>string</a>, <a class='existingWikiWord' href='/nlab/show/spinning+string'>spinning string</a>, <a class='existingWikiWord' href='/nlab/show/superstring'>superstring</a></p>
6640 </li>
6641
6642 <li>
6643 <p><a class='existingWikiWord' href='/nlab/show/membrane'>membrane</a></p>
6644 </li>
6645
6646 <li>
6647 <p><a class='existingWikiWord' href='/nlab/show/AKSZ+sigma-model'>AKSZ theory</a></p>
6648 </li>
6649 </ul>
6650 </li>
6651 </ul>
6652 </li>
6653 </ul>
6654 </li>
6655
6656 <li>
6657 <p><a class='existingWikiWord' href='/nlab/show/string+theory'>String Theory</a></p>
6658
6659 <ul>
6660 <li><a class='existingWikiWord' href='/nlab/show/string+theory+results+applied+elsewhere'>string theory results applied elsewhere</a></li>
6661 </ul>
6662 </li>
6663
6664 <li>
6665 <p><a class='existingWikiWord' href='/nlab/show/number+theory+and+physics'>number theory and physics</a></p>
6666
6667 <ul>
6668 <li><a class='existingWikiWord' href='/nlab/show/Riemann+hypothesis+and+physics'>Riemann hypothesis and physics</a></li>
6669 </ul>
6670 </li>
6671 </ul>
6672 <div>
6673 <p>
6674 <a href='/nlab/edit/physicscontents'>Edit this sidebar</a>
6675 </p>
6676 </div></div>
6677 </div>
6678 </div>
6679
6680 <h1 id='contents'>Contents</h1>
6681 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#references'>References</a></li></ul></div>
6682 <h2 id='idea'>Idea</h2>
6683
6684 <p>Our <a class='existingWikiWord' href='/nlab/show/galaxy'>galaxy</a>.</p>
6685
6686 <h2 id='references'>References</h2>
6687
6688 <ul>
6689 <li>Susan Gardner, Samuel D. McDermott, Brian Yanny, <em>The Milky Way, Coming into Focus: Precision Astrometry Probes its Evolution, and its Dark Matter</em> (<a href='https://arxiv.org/abs/2106.13284'>arXiv:2106.13284</a>)</li>
6690 </ul>
6691
6692 <p>See also</p>
6693
6694 <ul>
6695 <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Milky_Way'>Milky Way</a></em></li>
6696 </ul> </div>
6697 </content>
6698 </entry>
6699 </feed>
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