Higher categories of push–pull spans, I: construction and applications

Abstract 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 $$(\infty ,3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -categories "Equation missing", parametrized by an $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -category "Equation missing" with finite limits and a representable functor "Equation missing" with pushforwards, which contains correspondences in "Equation missing" with local systems in "Equation missing" that compose via a push–pull formula. We apply this general construction to provide an approximation "Equation missing" 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.