Homotopy hypothesis explained

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces.

One version of the hypothesis was claimed to be proved in the 1991 paper by Kapranov and Voevodsky.^[1] Their proof turned out to be flawed and their result in the form interpreted by Carlos Simpson is now known as the Simpson conjecture.^[2]

Formulations

There are many ways to formulate the hypothesis. For example, if we model our ∞-groupoids as Kan complexes (quasi-categories), 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

References

• John Baez, The Homotopy Hypothesis
• Book: q-alg/9705009 . 10.1007/BFb0026978 . An introduction to n-categories . Category Theory and Computer Science . Lecture Notes in Computer Science . 1997 . Baez . John C. . 1290 . 1–33 . 978-3-540-63455-3 .
• 2111.01000 . Grothendieck . Alexander . Pursuing Stacks . 2021 . math.CT .
• 10.1016/j.jpaa.2019.01.012 . The 2-dimensional stable homotopy hypothesis . 2019 . Gurski . Nick . Johnson . Niles . Osorno . Angélica M. . Journal of Pure and Applied Algebra . 223 . 10 . 4348–4383 . 1712.07218 .
• 10.1016/S0022-4049(02)00135-4 . Quasi-categories and Kan complexes . 2002 . Joyal . A. . Journal of Pure and Applied Algebra . 175 . 1–3 . 207–222 .
• Book: j.ctt7s47v . Higher Topos Theory (AM-170) . Lurie . Jacob . 2009 . Princeton University Press . 9780691140490 .
• Book: 10.1007/978-3-030-61524-6_2 . Joyal's Theorem, Applications, and Dwyer–Kan Localizations . Introduction to Infinity-Categories . Compact Textbooks in Mathematics . 2021 . Land . Markus . 97–161 . 978-3-030-61523-9. 1471.18001 .
• 1009.2331 . Maltsiniotis . Georges . Grothendieck inf ty -groupoids, and still another definition of inf ty -categories, §2.8. Grothendieck's conjecture (precise form). . 2010 . math.CT .
• Algebraic models for higher categories . 10.1016/j.indag.2010.12.004 . 2011 . Nikolaus . Thomas . Indagationes Mathematicae . 21 . 1–2 . 52–75 . 1003.1342 .
• 10.1090/noti2692 . Could ∞-Category Theory be Taught to Undergraduates? . 2023 . Riehl . Emily . Notices of the American Mathematical Society . 70 . 5 . 1 .
• alg-geom/9512006. 10.1023/A:1007747915317 . 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) . 1999 . Tamsamani . Zouhair . K-Theory . 16 . 51–99 .

Further reading

• 10.4171/JEMS/856 . A stratified homotopy hypothesis . 2018 . Ayala . David . Francis . John . Rozenblyum . Nick . Journal of the European Mathematical Society . 21 . 4 . 1071–1178 . 1502.01713 .

External links

• • Web site: What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? . MathOverflow.
• Jacob Lurie's Home Page
• Grothendieck's homotopy hypothesis . Simon . Henry . May 26, 2022. Grothendieck, a Multifarious Giant: Mathematics, Logic and Philosoph . https://www.chapman.edu/scst/conferences-and-events/grothendieck-conference.aspx.
• Web site: Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme . MathOverflow.

Notes and References

1. Kapranov . M. M. . Voevodsky . V. A. . inf ty -groupoids and homotopy types . Cahiers de Topologie et Géométrie Différentielle Catégoriques . 1991 . 32 . 1 . 29–46 . 1245-530X.
2. Simpson . Carlos . Homotopy types of strict 3-groupoids . 1998 . math/9810059.