A simplicial category for higher correspondences

TL;DR

This paper develops a simplicial categorical framework to classify inner fibrations and simplicial maps via higher correspondences, generalizing profunctors to the setting of $$-categories and using double category techniques.

Contribution

It introduces a simplicial category structure for higher correspondences, extending the classification of inner fibrations and simplicial maps in the context of $$-categories.

Findings

Classifies inner fibrations using $A$-indexed diagrams in a higher category.

Establishes a simplicial category structure for simplicial sets and correspondences.

Provides a framework for understanding higher morphisms as higher correspondences.

Abstract

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.