A functorial approach to $n$-abelian categories

TL;DR

This paper introduces a functorial framework for $n$-abelian categories, enabling the application of classical homological algebra techniques to higher homological algebra and generalizing key axioms from abelian categories.

Contribution

It reformulates $n$-abelian categories via finitely presented functors, extends axioms of abelian categories, and generalizes the higher Auslander correspondence.

Findings

Provides conditions for categories of modules to be $n$-abelian.

Extends the higher Auslander correspondence to $n$-abelian categories.

Describes when finitely generated projective modules form an $n$-abelian category.

Abstract

We develop a functorial approach to the study of $n$-abelian categories by reformulating their axioms in terms of their categories of finitely presented functors. Such an approach allows the use of classical homological algebra and representation theory techniques to understand higher homological algebra. As an application, we present two possible generalizations of the axioms "every monomorphism is a kernel" and "every epimorphism is a cokernel" of an abelian category to $n$-abelian categories. We also specialize our results to modules over rings, thereby describing when the category of finitely generated projective modules over a ring is $n$-abelian. Moreover, we establish a correspondence for $n$-abelian categories with additive generators, which extends the higher Auslander correspondence.