View source for Homotopy hypothesis

{{Short description|Hypothesis that the ∞-groupoids are equivalent to the topological spaces}}
In [[category theory]], a branch of mathematics, [[Grothendieck]]'s '''homotopy hypothesis''' states (very roughly speaking) that the [[∞-groupoid]]s are [[space (mathematics)|space]]s.

One version of the hypothesis was claimed to be proved in the 1991 paper by Kapranov and Voevodsky.<ref>{{cite journal |last1=Kapranov |first1=M. M. |last2=Voevodsky |first2=V. A. |title=<math>\infty</math>-groupoids and homotopy types |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=1991 |volume=32 |issue=1 |pages=29–46 |url=https://eudml.org/doc/91469 |issn=1245-530X}}</ref> Their proof turned out to be flawed and their result in the form interpreted by [[Carlos Simpson]] is now known as the [[Simpson conjecture]].<ref>{{cite arXiv |last1=Simpson |first1=Carlos |title=Homotopy types of strict 3-groupoids |date=1998 |eprint=math/9810059}}</ref>

== Formulations ==
There are many ways to formulate the hypothesis. For example, if we model our ∞-groupoids as [[Kan complex]]es ([[Quasi-category|quasi-categories]]<ref>{{harv|Land|2021|loc=2.1 Joyal’s Special Horn Lifting Theorem, Corollary 2.1.12}}</ref>), then the homotopy types of the geometric realizations of these sets give models for every homotopy type (perhaps in the weak form). It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types.

Depending on the definitions of ∞-groupoids, the hypothesis may trivially hold.

== See also ==
*''[[Pursuing Stacks]]''
*[[N-group (category theory)]]
*[[Quasi-category]]
*[[Algebraic homotopy]]

== Notes ==
{{reflist}}
== References ==
*John Baez, [http://math.ucr.edu/home/baez/homotopy/homotopy.pdf The Homotopy Hypothesis]
*{{cite book|arxiv=q-alg/9705009 |doi=10.1007/BFb0026978 |chapter=An introduction to n-categories |title=Category Theory and Computer Science |series=Lecture Notes in Computer Science |date=1997 |last1=Baez |first1=John C. |volume=1290 |pages=1–33 |isbn=978-3-540-63455-3 }}
*{{cite arXiv |eprint=2111.01000 |last1=Grothendieck |first1=Alexander |title=Pursuing Stacks |year=2021 |class=math.CT }}
*{{cite journal |doi=10.1016/j.jpaa.2019.01.012 |title=The 2-dimensional stable homotopy hypothesis |date=2019 |last1=Gurski |first1=Nick |last2=Johnson |first2=Niles |last3=Osorno |first3=Angélica M. |journal=Journal of Pure and Applied Algebra |volume=223 |issue=10 |pages=4348–4383 |arxiv=1712.07218 }}
*{{cite journal |doi=10.1016/S0022-4049(02)00135-4 |title=Quasi-categories and Kan complexes |date=2002 |last1=Joyal |first1=A. |journal=Journal of Pure and Applied Algebra |volume=175 |issue=1–3 |pages=207–222 }}
*{{cite book |jstor=j.ctt7s47v |title=Higher Topos Theory (AM-170) |last1=Lurie |first1=Jacob |year=2009 |publisher=Princeton University Press |isbn=9780691140490 }}
*{{cite book |doi=10.1007/978-3-030-61524-6_2 |chapter=Joyal's Theorem, Applications, and Dwyer–Kan Localizations |title=Introduction to Infinity-Categories |series=Compact Textbooks in Mathematics |date=2021 |last1=Land |first1=Markus |pages=97–161 |isbn=978-3-030-61523-9|zbl= 1471.18001 }}
*{{cite arXiv |eprint=1009.2331 |last1=Maltsiniotis |first1=Georges |title=Grothendieck <math>\infty</math>-groupoids, and still another definition of <math>\infty</math>-categories, §2.8. Grothendieck's conjecture (precise form). |date=2010 |class=math.CT }}
*{{cite journal |url=https://doi.org/10.1016/j.indag.2010.12.004 |doi=10.1016/j.indag.2010.12.004 |title=Algebraic models for higher categories |date=2011 |last1=Nikolaus |first1=Thomas |journal=Indagationes Mathematicae |volume=21 |issue=1–2 |pages=52–75 |arxiv=1003.1342 }}
*{{cite journal |doi=10.1090/noti2692 |title=Could ∞-Category Theory be Taught to Undergraduates? |date=2023 |last1=Riehl |first1=Emily |journal=Notices of the American Mathematical Society |volume=70 |issue=5 |page=1 }}
*{{cite journal 	|arxiv=alg-geom/9512006|doi=10.1023/A:1007747915317 |title=Sur des notions de ''n''-categorie et ''n''-groupoide non strictes via des ensembles multi-simpliciaux (On the notions of a nonstrict ''n''-category and ''n''-groupoid via multisimplicial sets) |date=1999 |last1=Tamsamani |first1=Zouhair |journal=K-Theory |volume=16 |pages=51–99 }}
== Further reading ==
*{{cite journal |doi=10.4171/JEMS/856 |title=A stratified homotopy hypothesis |date=2018 |last1=Ayala |first1=David |last2=Francis |first2=John |last3=Rozenblyum |first3=Nick |journal=Journal of the European Mathematical Society |volume=21 |issue=4 |pages=1071–1178 |arxiv=1502.01713 }}

== External links ==
*{{nlab|id=homotopy+hypothesis|title=homotopy hypothesis}}
*{{cite web |title=What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? |url=https://mathoverflow.net/q/234492 |website=MathOverflow}}
*[https://www.math.ias.edu/~lurie/ Jacob Lurie's Home Page]
*{{cite conference |url=https://www.chapman.edu/scst/conferences-and-events/grothendieck-files/henry-slides.pdf |title=Grothendieck's homotopy hypothesis |first=Simon |last=Henry |date=May 26, 2022|conference=Grothendieck, a Multifarious Giant: Mathematics, Logic and Philosoph |conference-url= https://www.chapman.edu/scst/conferences-and-events/grothendieck-conference.aspx}}
*{{cite web |title=Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme |url=https://mathoverflow.net/questions/266738/current-status-of-grothendiecks-homotopy-hypothesis-and-whiteheads-algebraic-h/ |website=MathOverflow}}

{{Category theory}}
{{Topology}}
[[Category:Homotopy theory]]
[[Category:Higher category theory]]
[[Category:Hypotheses]]
[[Category:Conjectures]]

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