Categorification of sheaf theory

TL;DR

This paper develops a systematic method for categorifying sheaf theory's six-functor formalism, enabling the construction of higher correspondences and applications like topological field theories.

Contribution

It introduces a procedure to produce compatible higher-level representations from a base correspondence representation, advancing the categorification of sheaf-theoretic formalisms.

Findings

Constructs a sequence of higher correspondence representations

Provides a general recipe for topological field theories

Advances the categorification of sheaf theory formalism

Abstract

We discuss a systematic procedure for categorifying presentable six-functor formalisms. Our main result produces, given the input of a representation of the $\infty$-category of correspondences of an $\infty$-category with finite limits $\mathcal{C}$, a compatible sequence of representations of the $(\infty,n)$-category of correspondences of $\mathcal{C}$ for every $n \geq 1$. As an application, we explain a general recipe for constructing topological field theories.