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 representable functor with pushforwards, which contains correspondences in with local systems in that 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.
