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, 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
References
- ^ Kapranov, M. M.; Voevodsky, V. A. (1991). "$\infty $-groupoids and homotopy types". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 32 (1): 29–46. ISSN 1245-530X.
- ^ Simpson, Carlos (1998). "Homotopy types of strict 3-groupoids". arxiv. arXiv:math/9810059.
- 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.
- Lurie, Jacob (2009). Higher Topos Theory (AM-170). Princeton University Press. ISBN 9780691140490. JSTOR j.ctt7s47v.
- Maltsiniotis, Georges (2010). "Grothendieck -groupoids, and still another definition of -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.
- 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.