Tensor structures on fibered categories

Abstract

Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased purely in terms of the associated external tensor product. More precisely, we construct a canonical dictionary relating the classical structures and properties of the internal tensor product to analogous structures and properties of the external tensor product: this applies to associativity, commutativity, and unit constraints, to projection formulae, as well as to monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories. For instance, we show how Mac Lane's classical pentagon and hexagon axioms can be stated using the external tensor product. Our results provide a satisfactory abstract framework to study monoidal structures in the setting of perverse sheaves.