  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.
