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Gysin functors, correspondences, and the Grothendieck-Witt category

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Daniel Dugger

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.

Keywords:
Burnside category, transfer map, Grothendieck-Witt ring, correspondences

2020 MSC:
18B10,55P91,12F10

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 6, pp 156-213.

Published 2022-01-19.

http://www.tac.mta.ca/tac/volumes/38/6/38-06.pdf

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