2024/01/24 by Luca Terenzi, Terenzi, Luca
Mathematics · #18D30 #18M05 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2401.13491
openalex publication_date 2024/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let S be a small category admitting binary products. We show that the whole theory of monoidal 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 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.