Gysin functors and the Grothendieck-Witt category, Part I

TL;DR

This paper introduces the Grothendieck-Witt category over a fixed ring, develops Gysin functors and correspondence categories, and demonstrates their relevance in contexts with symmetric monoidal structures and transfers.

Contribution

It defines the Grothendieck-Witt category, develops the theory of Gysin functors and correspondence categories, and proves a recognition theorem linking these structures to symmetric monoidal contexts.

Findings

Generalization of the Burnside category via correspondence categories

Recognition theorem for categories with symmetric monoidal structure and transfers

Examples illustrating the Grothendieck-Witt category structure

Abstract

We define the Grothendieck-Witt category over a fixed ground ring. In order to study the structure of this category, we introduce the general theory of Gysin functors and their associated categories of correspondences. The latter generalizes the familiar construction of the Burnside category over a finite group. We prove various results about the structure of these correspondence categories, and we prove a "recognition theorem" loosely saying that these correspondence categories naturally show up in situations where one has both a symmetric monoidal structure and transfers. Returning to the Grothendieck-Witt category, a few examples are worked out.