Speakers
Description
The Hochschild cohomology of a diagram of algebras, as introduced by Gerstenhaber and Schack, provides a natural framework for studying the cohomological properties of presheaves of algebras indexed by a small category. In this talk, we revisit and develop the connection between the Gerstenhaber--Schack complex and the Baues--Wirsching cohomology of categories, showing that a spectral sequence converging to diagrammatic Hochschild cohomology has its second page described by higher limits over the twisted arrow category.
A key simplification arises when restricting to diagrams of homological epimorphisms: the Baues--Wirsching cohomology reduces to classical functor cohomology over the indexing category, and the presence of a terminal object forces the spectral sequence to collapse. As a main application, we consider filtrations of finite simplicial complexes and the associated diagrams of incidence algebras. Exploiting the fact that injective simplicial maps induce surjective homological epimorphisms of incidence algebras, and the classical result that simplicial cohomology is Hochschild cohomology, we show that the diagrammatic Hochschild cohomology of such a filtration recovers the simplicial cohomology of the final complex.
The spectral sequence itself, however, carries richer information: its first page contains the classical persistent module of the filtration. At the same time, higher columns encode additional data whose geometric meaning remains an open question.