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


Igor Baković (Faculty of Natural Sciences and Mathematics, University of Split, Croatia)

Branislav Jurčo (Max Planck Institute for Mathematics, Bonn, Germany)


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.


bicategory, classifying topos, classifying space, principal bundle

2010 Mathematics Subject Classification

18D05, 18F20, 55U40

Published 1 January 2010