Gysin functors, correspondences, and the Grothendieck-Witt category

We introduce some general categorical machinery for studying Gysin functors (certain kinds of functors with transfers) and their associated categories of correspondences. These correspondence categories are closed, symmetric monoidal categories where all objects are self-dual. We also prove a limited reconstruction theorem: given such a closed, symmetric monoidal category (and some extra information) it is isomorphic to the correspondence category associated to a Gysin functor. Finally, if k is a field we use this technology to define and explore a new construction called the Grothendieck-Witt category of k.