  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.
