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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

We remark that the Graphoid intersection property, also called intersection property in Bayesian networks (Chapter 3 Theorem 1 [1]), is a particular case of an intersection property, in the sense of intersection of coverings, for factorisation spaces, also called factorisation models [2], factor graphs. Direct consequences of this are the equivalence between pairwise Markov property and local Markov property, the Hammersley-Clifford theorem.

This work studies operators mapping vector and scalar fields defined over a manifold M, and which commute with its group of diffeomorphisms Diff(M). We prove that in the case of scalar fields L p ω (M, R), those operators correspond to point-wise non-linearities, recovering and extending known results on R d . In the context of Neural Networks defined over M, it indicates that point-wise nonlinear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields L p ω (M, T M), we show that those operators are solely the scalar multiplication. It indicates that Diff(M) is too rich and that there is no universal class of nonlinear operators to motivate the design of Neural Networks over the symmetries of M.Preprint. Under review.