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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 1

### The classifying topos of a topological bicategory

Pages: 279 – 300

DOI: https://dx.doi.org/10.4310/HHA.2010.v12.n1.a14

#### Authors

#### Abstract

For any topological bicategory $\mathbb{B}$, the Duskin nerve $N\mathbb{B}$ of $\mathbb{B}$ is a simplicial space. We introduce the classifying topos $B\mathbb{B}$ of $\mathbb{B}$ as the Deligne topos of sheaves $Sh(N\mathbb{B})$ on the simplicial space $B\mathbb{B}$. It is shown that the category of geometric morphisms $Hom(Sh(X), B\mathbb{B}$) 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 $\mathbb{B}$-bundles. As a simple consequence, the geometric realization |$B\mathbb{B}$| of the nerve $B\mathbb{B}$ of a locally contractible topological bicategory $\mathbb{B}$ is the classifying space of principal $\mathbb{B}$-bundles, giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical $K$-theory. We also define classifying topoi of a topological bicategory $\mathbb{B}$ 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.

#### Keywords

bicategory; classifying topos; classifying space; principal bundle

#### 2010 Mathematics Subject Classification

18D05, 18F20, 55U40

Published 1 January 2010