A Simplicial Category for Higher Correspondences

In this work we propose a realization of Lurie’s prediction that inner fibrations \(p: X \rightarrow A\) are classified by A-indexed diagrams in a “higher category” whose objects are \(\infty \)-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and \(\infty \)-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.