## Algebraic & Geometric Topology

### Nerves and classifying spaces for bicategories

#### Abstract

This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory $C$. It proves the fact that the geometric realizations of all of these possible candidate “nerves of $C$” are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space $BC$ of the bicategory. Its other major result proves a direct extension of Thomason’s “Homotopy Colimit Theorem” to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the “Grothendieck construction on the diagram”. Our results provide coherence for all reasonable extensions to bicategories of Quillen’s definition of the “classifying space” of a category as the geometric realization of the category’s Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental “delooping” construction.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 1 (2010), 219-274.

Dates
Accepted: 8 November 2009
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882312

Digital Object Identifier
doi:10.2140/agt.2010.10.219

Mathematical Reviews number (MathSciNet)
MR2602835

Zentralblatt MATH identifier
1183.18005

#### Citation

Carrasco, Pilar; Cegarra, Antonio M; Garzón, Antonio R. Nerves and classifying spaces for bicategories. Algebr. Geom. Topol. 10 (2010), no. 1, 219--274. doi:10.2140/agt.2010.10.219. https://projecteuclid.org/euclid.agt/1513882312

#### References

• M Artin, B Mazur, On the van Kampen theorem, Topology 5 (1966) 179–189
• I Baković, Grothendieck construction for bicategories, Preprint, Available at http://www.irb.hr/korisnici/ibakovic/sgc.pdf
• J Bénabou, Introduction to bicategories, from: “Reports of the Midwest Category Seminar”, Springer, Berlin (1967) 1–77
• A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, Berlin (1972)
• L Breen, Théorie de Schreier supérieure, Ann. Sci. École Norm. Sup. $(4)$ 25 (1992) 465–514
• 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
• A M Cegarra, M Bullejos, A R Garzón, Higher-dimensional obstruction theory in algebraic categories, J. Pure Appl. Algebra 49 (1987) 43–102
• 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
• A M Cegarra, J Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set, Topology Appl. 153 (2005) 21–51
• A M Cegarra, J Remedios, The behaviour of the $\wwbar W$–construction on the homotopy theory of bisimplicial sets, Manuscripta Math. 124 (2007) 427–457
• J W Duskin, Simplicial matrices and the nerves of weak $n$–categories. I. Nerves of bicategories, Theory Appl. Categ. 9 (2001) 198–308 CT2000 Conference (Como)
• Z Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math. 106 (1984) 301–350
• R Garner, N Gurski, The low-dimensional structures that tricategories form, to appear in Mathematical Proc. Camb. Phil. Soc.
• P G Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25 (1982) 33–105
• P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag, 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 déscente, from: “Revêtements étales et groupe fondamental”, Lecture Notes in Math. 224, Springer, Berlin (1971) 145–194 Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1)
• N Gurski, An algebraic theory of tricategory, PhD thesis, University of Chicago (2007)
• N Gurski, Nerves of bicategories as stratified simplicial sets, J. Pure Appl. Algebra 213 (2009) 927–946
• V A Hinich, V V Schechtman, Geometry of a category of complexes and algebraic $K$–theory, Duke Math. J. 52 (1985) 399–430
• 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
• M M Kapranov, V A Voevodsky, $2$–categories and Zamolodchikov tetrahedra equations, from: “Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991)”, Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 177–259
• S Lack, ICONS
• S Lack, S Paoli, $2$–nerves for bicategories, $K$–Theory 38 (2008) 153–175
• S Mac Lane, Categories for the working mathematician, second edition, Graduate Texts in Math. 5, Springer, New York (1998)
• J P May, Simplicial objects in algebraic topology, Van Nostrand Math. Studies 11, D Van Nostrand Co., Princeton-Toronto-London (1967)
• J P May, The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer, Berlin (1972)
• J P May, Pairings of categories and spectra, J. Pure Appl. Algebra 19 (1980) 299–346
• I Moerdijk, J-A Svensson, Algebraic classification of equivariant homotopy $2$–types. I, J. Pure Appl. Algebra 89 (1993) 187–216
• D Quillen, Higher algebraic $K$–theory. I, from: “Algebraic $K$–theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)”, Lecture Notes in Math. 341, Springer, Berlin (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, Topology 13 (1974) 293–312
• C Simson, A closed model structure for $n$–categories, internal $hom$, $n$–staks and generalized Seifert–Van Kampen
• R Street, Two constructions on lax functors, Cahiers Topologie Géom. Différentielle 13 (1972) 217–264
• 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”, North-Holland, Amsterdam (1996) 529–577
• 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
• U Tillmann, Discrete models for the category of Riemann surfaces, Math. Proc. Cambridge Philos. Soc. 121 (1997) 39–49
• U Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997) 257–275
• K Worytkiewicz, K Hess, P E Parent, A Tonks, A model structure à la Thomason on 2-Cat, J. Pure Appl. Algebra 208 (2007) 205–236