# Indecomposables in the monomorphism category

The study of submodule categories is an old subject in representation theory going all the way back to beginning of the 20th century by work of Miller and Hilton. It has connections to, for example, Littlewood—Richardson tableaux, valuated p-groups and metabelian groups. In 2004 Ringel and Schmidmeier studied such categories using modern tools like Auslander—Reiten theory and covering theory.

A generalization of submodule categories, called (separated) monomorphism categories, has also been actively studied by several authors. They have found connections to for example cotorsion pairs, Gorenstein homological algebra, singularity theory and topological data analysis.

In this talk I will define submodule and monomorphism categories, and mention some of the known results about them. Then I will explain how they can be related to representations over stable categories via epivalences (also called representation equivalences), and how this can often be used to determine their indecomposables. I will also say something about our proof, which uses free monads on abelian categories. If time permits, I will discuss analogues of monomorphism categories for species. In particular, I will explain how our result can be used to give a characterization of Cohen-Macaulay finiteness for the algebras H associated to symmetrizable Cartan matrices introduced by Geiss-Leclerc-Schröer, assuming the terms in the symmetrizer are less than or equal to 2.

This is joint work with Nan Gao, Julian Külshammer and Chrysostomos Psaroudakis.