On the additive image of 0th persistent homology

I will present joint work with M. Botnan, S. Oppermann, and J. Steen on multiparameter persistent homology in degree zero. It is known that arbitrary diagrams of vector spaces and linear maps can be realized as homology of diagrams of simplicial complexes in some positive degree. We study the more restrictive case of degree zero, which corresponds to diagrams freely generated from sets and set maps. Despite their seemingly simple combinatorial nature, a full understanding of the structure of these representations remains elusive. I will summarize our findings and discuss some conjectures.