Higher Correspondences, Simplicial Maps and Inner Fibrations

Abstract

In this essay we propose a realization of Lurie's claim that inner fibrations $p: X \rightarrow C$ are classified by $C$-indexed diangrams 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. Correspondences between $\infty$-categories, and simplicial sets in general, are a generalization of the concept of profunctor (or bimodule) for categories. While categories, functors and profunctors are organized in a double category, we will exibit simplicial sets, simplicial maps, and correspondences as part of a simpliclal category. This allows us to make precise statements and proofs. Our main tool is the theory of double colimits.Comment: Questions, comments and corrections are welcom