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

TL;DR

This paper constructs a new framework for higher categories of correspondences with pushforwards, providing an approximation to Rozansky-Witten models within functorial field theory, advancing the understanding of their categorical structures.

Contribution

It introduces a symmetric monoidal $( rightarrow,3)$-category construction with pushforwards, approximating the conjectured 3-category of Rozansky-Witten models.

Findings

Constructs a symmetric monoidal $( rightarrow,3)$-category with pushforwards.

Provides an approximation to the 3-category of Rozansky-Witten models.

Relates to higher quasicoherent sheaves and functorial field theory.

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 $(\infty,3)$-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.