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On Multi-Determinant Functors for Triangulated Categories

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Ettore Aldrovandi and Cynthia Lester

We extend Deligne's notion of determinant functor to tensor triangulated categories.
Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and we provide a multicategorical version of the universal determinant functor for triangulated categories whose multiexactness properties are conveniently captured by a certain complex modeled by cubical shapes, which we introduce along the way. We then show that for a tensor triangulated category whose tensor admits a Verdier structure the resulting determinant functor takes values in a categorical ring.

Keywords:
Tensor triangulated category, determinant functor, multicategory, Picard groupoid, categorical ring, K-Theory, cubical complex

2020 MSC:
18G80, 18F25, 19D23, 18M65

*Theory and Applications of Categories,*
Vol. 39, 2023,
No. 27, pp 769-803.

Published 2023-09-06.

http://www.tac.mta.ca/tac/volumes/39/27/39-27.pdf

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