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 CHS22, we construct a family of symmetric monoidal (, 3) -categories PP (C; Q^) parametrized by an -category C with finite limits and a representable functor Q^ = C (-, Q^): Cᵒp CAlg (Cat₁) with pushforwards. We use this general construction and derived algebraic geometry to build ARW, an approximation to the 3-category of Rozansky-Witten models whose existence was conjectured by Kapustin-Rozansky KR10. In the second part we will study the dualizable objects of ARW and show that our construction extends the matrix factorization 2-category of Brunner-Carqueville-Roggenkamp BCR23.