Higher sheaf theory I: Correspondences

TL;DR

This paper establishes a universal property for the $( abla, n)$-category of correspondences, extending previous results and setting the stage for advanced applications in derived algebraic geometry.

Contribution

It generalizes the universal property of the category of correspondences to higher categories and provides conditions for extending functors to free cocompletions.

Findings

Proves a universal property for $( abla, n)$-categories of correspondences.

Provides conditions for functor extension to higher categories.

Lays groundwork for $( abla, n)$-categorical sheaf theories.

Abstract

We prove a universal property for the $(\infty, n)$-category of correspondences, generalizing and providing a new proof for the case $n = 2$ from [GR17]. We also provide conditions under which a functor out of a higher category of correspondences of $\mathcal{C}$ can be extended to a higher category of correspondences of the free cocompletion of $\mathcal{C}$. These results will be used in the sequels to this paper to construct $(\infty, n)$-categorical versions of the theories of quasicoherent and ind-coherent sheaves in derived algebraic geometry.