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.