Model Structures for Correspondences and Bifibrations

We study the notion of a bifibration in simplicial sets which generalizes the classical notion of two-sided discrete fibration studied in category theory. If $A$ and $B$ are simplicial sets we equip the category of simplicial sets over $A\times B$ with the structure of a model category for which the fibrant objects are the bifibrations from $A$ to $B$. We also equip the category of correspondences of simplicial sets from $A$ to $B$ with the structure of a model category. We describe several Quillen equivalences relating these model structure with the covariant model structure on the category of simplicial sets over $B^{\mathrm{op}}\times A$.