On Multi-Determinant Functors for Triangulated Categories

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.


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