Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras

In this paper, we study the Hochschild cohomology of diagrams of algebras introduced by Gerstenhaber and Schack and provide computations for filtrations of incidence algebras. Our aims are threefold: firstly, we revisit and explore the connection between the Gerstenhaber-Schack complexes and the Baues-Wirsching cohomology of categories. Secondly, we analyse the behaviour of (diagrammatic) Hochschild cohomology in the context of homological epimorphisms of algebras. Thirdly, we study diagrams of incidence algebras. In particular, as a main application, we compute the diagrammatic Hochschild cohomology of diagrams of incidence algebras associated to finite filtrations of simplicial complexes.