Homotopy hypothesis

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^[3]), 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

1. Kapranov, M. M.; Voevodsky, V. A. (1991). "${\displaystyle \infty }$-groupoids and homotopy types". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 32 (1): 29–46. ISSN 1245-530X.
2. Simpson, Carlos (1998). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.
3. (Land 2021, 2.1 Joyal’s Special Horn Lifting Theorem, Corollary 2.1.12)

References

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

Further reading

• Ayala, David; Francis, John; Rozenblyum, Nick (2018). "A stratified homotopy hypothesis". Journal of the European Mathematical Society. 21 (4): 1071–1178. arXiv:1502.01713. doi:10.4171/JEMS/856.

External links

• homotopy hypothesis at the nLab
• "What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?". MathOverflow.
• Jacob Lurie's Home Page
• Henry, Simon (May 26, 2022). Grothendieck's homotopy hypothesis (PDF). Grothendieck, a Multifarious Giant: Mathematics, Logic and Philosoph.
• "Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme". MathOverflow.