  We introduce an original notion of extra-fine sheaf on a topological space,
and a variant (hyper-extra-fine) for which \v{C}ech 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 Mat\'u\v{s}). We finish with a
comparison theorem between the \v{C}ech 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 \v{C}ech
cohomology, and their application to the marginal problem.
