Gysin functors and the Grothendieck-Witt category, Part I

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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.