Categorification of sheaf theory

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