Higher categories of push-pull spans, I: Construction and applications

This is the first part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. Motivated by work of Calaque-Haugseng-Scheimbauer, we construct a family of symmetric monoidal $(\infty,3)$-categories parametrized by an $\infty$-category with finite limits and a functor into symmetric monoidal $\infty$-categories, such that the functor admits pushforwards. This $(\infty,3)$-category contains correspondences in the base $\infty$-category equipped with local systems, which compose via a push-pull formula. We apply this general construction to provide an approximation to the $3$-category of Rozansky-Witten models whose existence was conjectured by Kapustin-Rozansky-Saulina; this approximation behaves like a "commutative" version of the conjectured $3$-category and is related to work of Stefanich on higher quasicoherent sheaves.