Abstract
In this paper we deal with the simplicial nerve (Street’s geometric nerve) of bicategories and tricategories. We prove that if 𝔸 is a trigroupoid then Ner(𝔸) is a Kan simplicial set which moreover turns out to be a (Duskin-Glenn) 3-hypergroupoid, generalizing in this way, the analogous result for the simplicial nerve of bigroupoids, due to Duskin. We associate to any Kan simplicial set X, its homotopy bigroupoid Π2 X, and its homotopy trigroupoid Π3 X. We prove that there exists a simplicial map v:X→Ner(Π2 X) which is a surjective weak 2-equivalence and an isomorphism if X is a Kan 2-hypergroupoid, giving another proof of Duskin’s characterization for the simplicial nerve of bigroupoids. In the 3-dimensional case, there also exists a simplicial map v : X → Ner(Π3 X) that is a weak 3-equivalence but, in this case, not necessarily surjective, not even under the hypergroupoid hypothesis. As a corollary, we conclude that any Kan simplicial set with trivial homotopy groups at dimensions ≥ 4 is homotopy equivalent to the nerve of a trigroupoid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bénabou J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar. Lecture Notes in Math, vol. 47, pp. 1–77. Springer-Verlag, Berlin-New York (1967)
Berger, C.: Double loop spaces, braided monoidal categories and algebraic 3-type of space. Contemp. Math. 227, 49–66 (1999)
Carrasco, P., Cegarra, A.M.: (Braided) tensor structures on homotopy groups and nerves of (braided) categorical groups. Comm. Algebra 24(13), 3995–4058 (1996)
Carrasco, P., Cegarra, A.M., Garzón, A.R.: Nerves and classifying spaces of bicategories. Alg. Geom. 10, 219–274 (2010)
Carrasco, P., Cegarra, A.M., Garzón, A.R.: Classifyng spaces for braided monoidal categories and lax diagrams of bicategories. Adv. Math. 226, 419–483 (2011)
Cegarra, A.M., Heredia, B.A.: Geometric realizations of tricategories. arXiv:1203.3664v2 (2012)
Curtis, E.B.: Simplicial homotopy theory. Adv. Math. 6, 107–209 (1971)
Duskin, J.: Simplicial methods and the interpretation of triple cohomology. Memoir Amer. Math. Soc. 3(2), 1–135 (1975)
Duskin, J.: Simplicial matrices and the nerves of weak n-categories I: Nerves of bicategories. Theory Appl. Cat. 9(10), 198–308 (2002)
Glenn, P.: Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Algebra 25, 33–107 (1982)
Goers, P.G., Jardine, J.F.: Simplicial homotopy theory. Progress in Math, vol. 174. Birkhäuser Verlag (1999)
Gordon, R., Power, A.J., Street, R.: Coherence for tricategories. Mem. Amer. Math. Soc. 117(558) (1995)
Gurski, N.: An algebraic theory of tricategories. University of Chicago, PhD thesis (2006)
Gurski, N.: Nerves of bicategories as stratified simplicial sets. J. Pure Appl. Algebra 213, 927–946 (2009)
Kan, D.M.: A combinatorial definition of the homotopy groups. Ann. Math. 67(2), 282–312 (1958)
Kelly, G.M., Street, R.: Review of the elements of 2-categories. In: Category Seminar. Lecture Notes in Math, vol. 420, pp. 75–103. Springer, Heidelberg (1974)
May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand, Princeton (1967)
Street, R.: The algebra of oriented simplixes. J. Pure Appl. Algebra 49, 283–335 (1987)
Street, R.: Categorical structures. In: Handbook of Algebra 1, pp. 529–577 North-Holland (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carrasco, P. Nerves of Trigroupoids as Duskin-Glenn’s 3-Hypergroupoids. Appl Categor Struct 23, 673–707 (2015). https://doi.org/10.1007/s10485-014-9374-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-014-9374-7