The classifying topos of a topological bicategory

TL;DR

This paper introduces the classifying topos of a topological bicategory, linking sheaf categories on its nerve to principal bundles, and explores various nerve constructions and their properties.

Contribution

It defines the classifying topos for topological bicategories and establishes its equivalence with principal bundle categories, extending classical results to bicategorical contexts.

Findings

The classifying topos of a topological bicategory is equivalent to sheaves on its nerve.

The geometric realization of the nerve classifies principal bicategory bundles.

Different nerve constructions yield related classifying topoi with specific properties.

Abstract

For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.