An introduction to monomorphism categories

TL;DR

This paper introduces monomorphism categories, surveys recent theoretical results, presents a new classification over cyclic abelian groups, and discusses their relationships and complexity, including a correction of previous literature.

Contribution

It provides a comprehensive introduction, recent advances, a new classification result, and clarifies the wildness of certain monomorphism categories.

Findings

Classification of monomorphism categories of finite type over cyclic abelian groups

Establishment of the wildness of a specific monomorphism category

Connections between submodule categories and p-valuated abelian groups

Abstract

This manuscript was written for the Proceedings of the ICRA 2022 in Buenos Aires. It can be divided into four parts: The first part is an introduction to the theory of monomorphism categories, including a short survey on some representation theoretic results. The second part is a summary of some recent results on monomorphism categories based on joint work with Nan Gao, Julian K\"ulshammer and Chrysostomos Psaroudakis. It also includes a new result, namely a classification of all monomorphism categories of finite type over cyclic abelian groups. The third part concerns the relationship between submodule categories and $p$-valuated abelian groups. The last part contains a proof of wildness of a certain monomorphism category, rectifying a statement in the literature.