engleski

The classifying topos of a topological bicategory

For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space 2NC. It is shown that the category of topos morphisms from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos Sh(N2C) is naturally equivalent to the category of principal 2C-bundles. As a simple consequence, the geometric realization of the nerve N2C of a locally contractible topological bicategory 2C is the classifying space of principal 2C-bundles (on CW complexes), giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory 2C 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; Duskin nerve; classifying topos; bisimplicial space

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