2-fold structures and homotopy theory

It is well-known that correspondences between categories, also known as profunctors, serve in classifying functors. More precisely, every functor F : X → A straightens into a lax mapping χF : A → Catprof from A into a 2-category of categories and profunctors ([45]). We give a conceptual treatment of this fact from the lens of double category theory, contending the latter to be most natural environment to express this result. Then we venture into the world of simplicial sets and prove an analogous theorem. The notion of correspondence is easy to extend to simplicial sets, but a suitable double category may not be formed due to the lack of a natural tensor product. Nonetheless, we show that there is a natural simplicial category structure once we invoke higher correspondences. In proving our result we extend some notions from double category theory into the world of simplicial categories. As an application we obtain a realization of Lurie’s prediction that inner fibrations are classified by mappings into a higher category of correspondences between ∞-categories.