Functors and morphisms determined by subcategories

TL;DR

This paper investigates the existence and uniqueness of minimal right determiners in certain categories, extending known formulas and providing new ones for specific representation categories.

Contribution

It extends the Auslander-Reiten-Smal{ o}-Ringel formula to Hom-finite hereditary abelian categories with enough projectives and derives a new formula for finitely presented representations of certain quivers.

Findings

The Auslander-Reiten-Smal{ o}-Ringel formula holds in the specified categories.

A new formula for minimal right determiners in finitely presented quiver representations.

Confirmation of existence and uniqueness conditions for minimal right determiners.

Abstract

We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smal{\o}-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.