Homology, Homotopy and Applications

The classifying topos of a topological bicategory

Igor Baković and Branislav Jurčo

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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.

Article information

Homology Homotopy Appl., Volume 12, Number 1 (2010), 279-300.

First available in Project Euclid: 28 January 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D05: Double categories, 2-categories, bicategories and generalizations 18F20: Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30] 55U40: Topological categories, foundations of homotopy theory

Bicategory classifying topos classifying space principal bundle


Baković, Igor; Jurčo, Branislav. The classifying topos of a topological bicategory. Homology Homotopy Appl. 12 (2010), no. 1, 279--300. https://projecteuclid.org/euclid.hha/1296223830

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