Computing relative Betti diagrams of functors indexed by posets

One of the fundamental results of topological data analysis is that functors from the real line (viewed as a posetal category) to the category of finite-dimensional vector spaces split as the direct sum of simple interval modules. When replacing the real line with an arbitrary (finite) poset, this decomposition no longer holds. Instead, we turn to relative homological algebra to develop new invariants for such functors. We focus on relative projective resolutions, which are exact sequences of functors that are projective relative to a given family of “simple" modules. Under certain reasonable conditions, these resolutions consist of direct sums of simple functors, whose multiplicities we call the relative Betti diagrams. We then compute these relative Betti diagrams using local Koszul complexes, which is simpler than first computing the full resolution.

After presenting our general results, I will focus on the case of Betti diagrams relative to lower hook modules, which are closely related to signed barcodes (Botnan-Oppermann-Oudot 2024), and explore some tricks to improve their computational complexity. In particular, I will present a spectral sequence that converts kernels into cokernels, the latter being easier to handle computationally.

This is joint work with Wojciech Chachólski, Andrea Guidolin, Martina Scolamiero, and Francesca Tombari.