The classifying topos of a topological bicategory

Igor Baković ; Branislav Jurčo

Homology, homotopy, and applications, Tome 12 (2010) no. 1, pp. 279-300.

Résumé

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.

Détail

DOI : 10.4310/HHA.2010.v12.n1.a14

Classification : 18D05, 18F20, 55U40
Keywords: bicategory, classifying topos, classifying space, principal bundle