2024/01/24 by Luca Terenzi, Terenzi, Luca
Mathematics · #18D30 #18M05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2401.13489
openalex publication_date 2024/01/24 · openalex created_date 2024/01/26 · openalex updated_date 2026/07/28
Let S be a small category, and suppose that we are given two (non-full) subcategories Ssm and Scl that generate all morphisms of S under composition in the same way as morphisms of quasi-projective algebraic varieties are generated by smooth morphisms and closed immersions. We show that a monoidal structure on a given S-fibered category is completely determined by its restrictions to Ssm and Scl; in fact, any such pair of monoidal structures satisfying a natural coherence condition uniquely determines a monoidal structure over S. The same principle applies to morphisms of S-fibered categories and monoidality thereof. Under further assumptions on the subcategories Ssm and Scl, and with suitable restrictions on the S-fibered categories and morphisms involved, we provide a variant of the above factorization method in which inverse images under closed immersion are partially replaced by the corresponding direct images: the latter variant is more adapted to the setting of perverse sheaves.