Geometric Realizations of Tricategories
Antonio M. Cegarra, Benjam\'in A. Heredia

TL;DR
This paper investigates the relationships among different nerve constructions of tricategories and proves their geometric realizations are homotopy equivalent, providing a unified framework for classifying spaces and extending classical results.
Contribution
It establishes homotopy equivalences among various nerves of tricategories and extends classical results on classifying spaces and loop spaces to tricategorical contexts.
Findings
Homotopy equivalence of geometric realizations of different tricategory nerves.
Extension of classifying space properties to tricategories.
Connection between bicategorical groups and connected homotopy 3-types.
Abstract
Any tricategory characteristically has associated various simplicial or pseudo-simplicial objects. This paper explores the relationship amongst three of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the tricategory, the simplicial bicategory termed its Segal nerve, and the simplicial set called its Street geometric nerve, and it proves the fact that the geometric realizations of all of these possible candidate 'nerves of the tricategory' are homotopy equivalent. Our results provide coherence for all reasonable extensions to tricategories of Quillen's definition of the 'classifying space' of a category as the geometric realization of the category's Grothendieck nerve. Many properties of the classifying space construction for tricategories may be easier to establish depending on the nerve used for realizations. For instance, by using Grothendieck nerves we state…
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