The classifying topos of a topological bicategory

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

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Published 1 January 2010