Extra-fine sheaves and interaction decompositions

Abstract

We introduce an original notion of extra-fine sheaf on a topological space, for which Čech cohomology in strictly positive dimension 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. Contents 1. Introduction 1 2. Fine, extra-fine, super-local and acyclic 5 3. Alexandrov topologies and sheaves 9 4. Interaction decomposition 12 5. Factorization of free sheaves 18 6. Nerves of categories and nerves of coverings 23 Appendix A. Topology and sheaves 37 Appendix B. Čech cohomology 39 Appendix C. Finite probability functors 40 References 44

On Information Links

“…Multivariate mutual informations I k appears in this context as coboundaries [5,4], and quantify refined and local statistical dependences in the sens that n variables are mutually independent if and only if all the I k vanish (with 1 < k < n, giving (2 n − n − 1) functions), whereas the Total Correlations G k quantify global or total dependences, in the sens that n variables are mutually independent if and only if G k = 0 (theorem 2 [6]). As preliminary uncovered by several related studies, information functions and statistical structures not only present some co-homological but also homotopical features that are finer invariants [4,7,21]. Notably, proposition 9 in [6], underlines a correspondence of the minima I 3 = −1 of the mutual information between 3 binary variables with Borromean link.…”

On Information (pseudo) Metric

“…Linear and convex combinations of random variables are studied in the context of information homotopy to be submitted [5]. Moreover without proof, we expect that the topology induced by this metric to be the poset topology also called Alexandrov topology corresponding to partition poset and as suggested by the work of Bennequin et al [10].…”