A persistent variation: data field theory via sheaves of incidence algebras

We will extend persistent (co-)homology by considering diagrams of incidence algebras of finite posets. By using results of Gerstenhaber et al. we will show what is the link between diagram cohomology, Hochschild cohomology, and multidimensional persistence connecting seemingly uncorrelated research areas as topological data analysis, noncommutative geometry and topological field theory (via Cattaneo - Mnev - Reshetikhin work on cellular field theory).