Bicategorical homotopy pullbacks

TL;DR

This paper extends Quillen's Theorem B to bicategories, providing a bicategory-theoretical interpretation of homotopy-fibre products, with applications to monoidal categories and crossed modules.

Contribution

It presents a bicategory-theoretical extension of Quillen's Theorem B, linking bicategories with homotopy pullbacks of classifying spaces.

Findings

Extended Quillen's Theorem B to bicategories

Provided a bicategory interpretation of homotopy-fibre products

Applied results to monoidal categories and crossed modules

Abstract

The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between B\'enabou's bicategories and the homotopy types of their classifying spaces. Mainly, we state and prove an extension of Quillen's Theorem B by showing, under reasonable necessary conditions, a bicategory-theoretical interpretation of the homotopy-fibre product of the continuous maps induced on classifying spaces by a diagram of bicategories $\mathcal{A}\to\mathcal{B}\leftarrow \mathcal{A}'$. Applications are given for the study of homotopy pullbacks of monoidal categories and of crossed modules.