Algebraic & Geometric Topology

Comparing geometric realizations of tricategories

Antonio M Cegarra and Benjamín A Heredia

Full-text: Open access


This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories. Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve. We prove that the geometric realizations of all of these ‘nerves of the tricategory’ are homotopy equivalent. By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space. With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3–types.

Article information

Algebr. Geom. Topol., Volume 14, Number 4 (2014), 1997-2064.

Received: 4 September 2013
Revised: 5 December 2013
Accepted: 6 December 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D05: Double categories, 2-categories, bicategories and generalizations 55P15: Classification of homotopy type
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 55P35: Loop spaces

monoidal bicategory tricategory nerve classifying space homotopy type loop space


Cegarra, Antonio M; Heredia, Benjamín A. Comparing geometric realizations of tricategories. Algebr. Geom. Topol. 14 (2014), no. 4, 1997--2064. doi:10.2140/agt.2014.14.1997.

Export citation


  • J C Baez, The homotopy hypothesis Available at \setbox0\makeatletter\@url {\unhbox0
  • J C Baez, M Neuchl, Higher-dimensional algebra, I: Braided monoidal $2$–categories, Adv. Math. 121 (1996) 196–244
  • C Balteanu, Z Fiedorowicz, R Schwänzl, R Vogt, Iterated monoidal categories, Adv. Math. 176 (2003) 277–349
  • J Bénabou, Introduction to bicategories, from: “Reports of the Midwest Category Seminar”, Springer (1967) 1–77
  • C Berger, Double loop spaces, braided monoidal categories and algebraic $3$–type of space, from: “Higher homotopy structures in topology and mathematical physics”, Contemp. Math. 227, Amer. Math. Soc. (1999) 49–66
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
  • M Bullejos, A M Cegarra, On the geometry of $2$–categories and their classifying spaces, $K\!$–Theory 29 (2003) 211–229
  • M Bullejos, A M Cegarra, Classifying spaces for monoidal categories through geometric nerves, Canad. Math. Bull. 47 (2004) 321–331
  • P Carrasco, A M Cegarra, (Braided) tensor structures on homotopy groupoids and nerves of (braided) categorical groups, Comm. Algebra 24 (1996) 3995–4058
  • P Carrasco, A M Cegarra, Schreier theory for central extensions of categorical groups, Comm. Algebra 24 (1996) 4059–4112
  • P Carrasco, A M Cegarra, A R Garzón, Nerves and classifying spaces for bicategories, Algebr. Geom. Topol. 10 (2010) 219–274
  • P Carrasco, A M Cegarra, A R Garzón, Classifying spaces for braided monoidal categories and lax diagrams of bicategories, Adv. Math. 226 (2011) 419–483
  • A M Cegarra, A R Garzón, Homotopy classification of categorical torsors, Appl. Categ. Structures 9 (2001) 465–496
  • A M Cegarra, E Khmaladze, Homotopy classification of graded Picard categories, Adv. Math. 213 (2007) 644–686
  • E Cheng, N Gurski, The periodic table of $n$–categories for low dimensions II: Degenerate tricategories
  • J W Duskin, Simplicial matrices and the nerves of weak $n$–categories. I: Nerves of bicategories, Theory Appl. Categ. 9 (2002) 198–308
  • Z Fiedorowicz, The symmetric bar construction (1998) Available at \setbox0\makeatletter\@url {\unhbox0
  • R Garner, N Gurski, The low-dimensional structures formed by tricategories, Math. Proc. Cambridge Philos. Soc. 146 (2009) 551–589
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • R Gordon, A J Power, R Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995) vi+81
  • A Grothendieck, Catégories fibrées et descente, from: “Séminaire de géométrie algébrique du Bois Marie 1966/67”, (P Berthelot, A Grothendieck, L Illusie, editors), Lecture Notes in Math. 224, Springer (1971) 145–194
  • N Gurski, Nerves of bicategories as stratified simplicial sets, J. Pure Appl. Algebra 213 (2009) 927–946
  • N Gurski, Loop spaces, and coherence for monoidal and braided monoidal bicategories, Adv. Math. 226 (2011) 4225–4265
  • N Gurski, Biequivalences in tricategories, Theory Appl. Categ. 26 (2012) 349–384
  • N Gurski, Coherence in three-dimensional category theory, Cambridge Tracts in Mathematics 201, Cambridge Univ. Press (2013)
  • K A Hardie, K H Kamps, R W Kieboom, A homotopy bigroupoid of a topological space, Appl. Categ. Structures 9 (2001) 311–327
  • L Illusie, Complexe cotangent et déformations, II, Lecture Notes in Mathematics 283, Springer (1972)
  • J F Jardine, Supercoherence, J. Pure Appl. Algebra 75 (1991) 103–194
  • A Joyal, R Street, Braided tensor categories, Adv. Math. 102 (1993) 20–78
  • A Joyal, M Tierney, Algebraic homotopy types, Handwritten lecture notes (1984)
  • M M Kapranov, V A Voevodsky, $2$–categories and Zamolodchikov tetrahedra equations, from: “Algebraic groups and their generalizations: quantum and infinite-dimensional methods”, (W J Haboush, B J Parshall, editors), Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 177–259
  • S Lack, Icons, Appl. Categ. Structures 18 (2010) 289–307
  • S Lack, A Quillen model structure for Gray-categories, J. $K\!$–Theory 8 (2011) 183–221
  • S Lack, S Paoli, $2$–nerves for bicategories, $K\!$–Th. 38 (2008) 153–175
  • O Leroy, Sur une notion de $3$–catégorie adaptée à l'homotopie (1994)
  • J P May, The spectra associated to permutative categories, Topology 17 (1978) 225–228
  • D Quillen, Higher algebraic $K\!$–theory, I, from: “Algebraic $K\!$–theory, I: Higher $K\!$–theories”, Lecture Notes in Math. 341, Springer (1973) 85–147
  • G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105–112
  • G Segal, Categories and cohomology theories, Topol. 13 (1974) 293–312
  • C Simpson, Homotopy theory of higher categories, New Mathematical Monographs 19, Cambridge Univ. Press (2012)
  • J D Stasheff, Homotopy associativity of $H$–spaces, I, II, Trans. Amer. Math. Soc. 108 (1963) 293–312
  • D Stevenson, The geometry of bundle gerbes, PhD thesis, University of Adelaide (2000) Available at \setbox0\makeatletter\@url \unhbox0
  • R Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283–335
  • R Street, Categorical structures, from: “Handbook of algebra, Vol. 1”, (M Hazewinkel, editor), Handb. Algebr. 1, Elsevier/North-Holland, Amsterdam (1996) 529–577
  • R Street, Categorical and combinatorial aspects of descent theory, Appl. Categ. Structures 12 (2004) 537–576
  • Z Tamsamani, Sur des notions de $n$–catégorie et $n$–groupoï de non strictes via des ensembles multi-simpliciaux, $K\!$–Theory 16 (1999) 51–99
  • R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109