  Motivated by various developments in algebraic combinatorics and its
applications, we investigate here the fine structure of a fundamental but
little known theorem, the Gerstenhaber and Schack cohomology comparison
theorem.The theorem classically asserts that there is a cochain equivalence
between the usual singular cochain complex of a simplicial complex and the
relative Hochschild complex of its incidence algebra, and a quasi-isomorphism
with the standard Hochschild complex. Here, we will be mostly interested in its
application to arbitrary posets (or, equivalently, finite topologies) and their
incidence algebras. We construct various structures, classical and new, on the
above two complexes: cosimplicial, differential graded algebra, operadic and
brace algebra structures and show that the comparison theorem preserves all of
them. These results provide non standard insights on links between the theory
of posets, incidence algebras, endomorphism operads and finite and
combinatorial topology. By non standard, we refer here to the use of relative
versions of Hochschild complexes and operads.
