Algebraic & Geometric Topology

Comparing geometric realizations of tricategories

Antonio M Cegarra and Benjamín A Heredia

Full-text: Open access

Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.agt/1513715957

Digital Object Identifier
doi:10.2140/agt.2014.14.1997

Mathematical Reviews number (MathSciNet)
MR3331608

Zentralblatt MATH identifier
1321.18004

Subjects
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

Keywords
monoidal bicategory tricategory nerve classifying space homotopy type loop space

Citation

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. https://projecteuclid.org/euclid.agt/1513715957


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