Posets, their incidence algebras and relative operads, and the cohomology comparison theorem

Abstract

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 topological spaces satisfying the $T_0$ separation axiom) 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.