The classifying topos of a topological bicategory

We introduce the classifying topos BB of a topological bicategory B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB, the (Duskin) nerve of the bicategory B. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos BB is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. Similar construction works also for other types of nerves of the bicategory B (e.g., the nerves introduced by Lack and Paoli or Simpson and Tamsamani).