From Morphism Categories to Functor Categories

TL;DR

This paper explores the relationship between morphism categories and functor categories in representation theory, establishing functorial connections that preserve key properties and applying these to Auslander algebras and stable Auslander-Reiten quivers.

Contribution

It constructs a functor between morphism and functor categories that preserves almost split sequences and applies this to auto-equivalences and Auslander-Reiten theory.

Findings

Many almost split sequences are preserved by the functor.

Established auto-equivalences in wide subcategories of module categories.

Analyzed Auslander-Reiten translates of simple modules in finite type algebras.

Abstract

For a nice-enough category $\mathcal{C}$, we construct both the morphism category ${\rm H}(\mathcal{C})$ of $\mathcal{C}$ and the category ${\rm mod}\mbox{-}\mathcal{C}$ of all finitely presented contravariant additive functors over $\mathcal{C}$ with values in Abelian groups. The main theme of this paper, is to translate some representation-theoretic attributes back and forth from one category to the other. This process is done by using an appropriate functor between these two categories, an approach which seems quite promising in particular when we show that many of almost split sequences are preserved by this functor. We apply our results to the case of wide subcategories of module categories to obtain certain auto-equivalences over them. Another part of the paper deals with Auslander algebras arising from algebras of finite representation type. In fact, we apply our results to study the Auslander-Reiten translates of simple modules over such algebras. In the last parts, we try to recognize particular components in the stable Auslander-Reiten quiver of Auslander algebras arising from self-injective algebras of finite representation type.