2020/09/26 by Daniel Bennequin, Bennequin, Daniel, Olivier Peltre +5
Computer Science · Mathematics · #55N30 #55U10 Secondary 06A11 #60B99 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 14F05 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2009.12646
openalex publication_date 2020/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce an original notion of extra-fine sheaf on a topological space, and a variant (hyper-extra-fine) for which Čech cohomology in strictly positive degree vanishes. We provide a characterization of such sheaves when the topological space is a partially ordered set (poset) equipped with the Alexandrov topology. Then we further specialize our results to some sheaves of vector spaces and injective maps, where extra-fineness is (essentially) equivalent to the decomposition of the sheaf into a direct sum of subfunctors, known as interaction decomposition, and can be expressed by a sum-intersection condition. We use these results to compute the dimension of the space of global sections when the presheaves are freely generated over a functor of sets, generalizing classical counting formulae for the number of solutions of the linearized marginal problem (Kellerer and Matúš). We finish with a comparison theorem between the Čech cohomology associated to a covering and the topos cohomology of the poset with coefficients in the presheaf, which is also the cohomology of a cosimplicial local system over the nerve of the poset. For that, we give a detailed treatment of cosimplicial local systems on simplicial sets. The appendixes present presheaves, sheaves and Čech cohomology, and their application to the marginal problem.