Classifying spaces for braided monoidal categories and lax diagrams of bicategories

TL;DR

This paper explores the relationship between higher categorical structures and their classifying spaces, showing how lax diagrams of bicategories can model homotopy types and produce explicit models for braided monoidal categories.

Contribution

It proves that the classifying space of a bicategory from a lax diagram is a homotopy colimit of component spaces, and constructs explicit models for braided monoidal categories using geometric nerves.

Findings

Classifying space of a bicategory from a lax diagram is a homotopy colimit.

Double delooping spaces match the homotopy type of the underlying category.

Explicit simplicial models for braided monoidal categories are constructed.

Abstract

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.