The classifying topos of a topological bicategory

Abstract

For any topological bicategory $\mathbb{B}$, the Duskin nerve $N\mathbb{B}$ of $B$ is a simplicial space. We introduce the classifying topos $\mathcal{B}\mathbb{B}$ of $\mathbb{B}$ as the Deligne topos of sheaves Sh$(N\mathbb{B})$ on the simplicial space $N\mathbb{B}$. It is shown that the category of geometric morphisms Hom(Sh($X), \mathcal{B}\mathbb{B}$) 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 $|N\mathbb{B}|$ of the nerve $N\mathbb{B}$ 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.