The aim of the talk is to describe the weak categorical quiver minor theorem. We will introduce the framework of quasi-Groebner categories, as developed by Sam and Snowden, and use it to study structural properties (e.g. bound on ranks and order of torsion) of graph homologies, in the spirit of Miyata, Proudfoot and Ramos. More specifically, we will focus on magnitude (co)homology, as introduced by Hepworth and Willerton, and we will show that magnitude cohomology yields finitely generated functors on the category of directed graphs with bounded genus. Then, we will discuss some main applications. This is joint work with Carlo Collari and Eric Ramos.