In topological data analysis, two-parameter persistence can be studied using the
representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of
type A. In a previous work, we defined interval approximations using restrictions to essential
vertices of intervals together with Mobius inversion. In this work, we consider homological
approximations using interval resolutions, and show that the interval resolution global
dimension is finite for finite posets and that it is equal to the maximum of the interval …

In topological data analysis, in contrast to the case of one-parameter persistent homology,
two-parameter persistent homology presents algebraic difficulties due to its wild
representation type. We consider approximations of two-parameter persistence modules:(1)
In a previous work, we defined interval approximations using “compression” to essential
vertices of intervals together with Möbius inversion.(2) Another idea is to consider
homological approximations of persistence modules using interval resolutions. In this work …