$n$-abelian and $n$-exact categories

TL;DR

This paper introduces $n$-abelian and $n$-exact categories as higher homological algebra analogs of classical categories, establishing their properties, examples, and connections to existing structures.

Contribution

It defines $n$-abelian and $n$-exact categories, relates them to $n$-cluster-tilting subcategories, and explores their structures and examples in various mathematical contexts.

Findings

$n$-cluster-tilting subcategories are $n$-abelian or $n$-exact.

$n$-abelian categories can be realized as $n$-cluster-tilting subcategories.

Stable categories of Frobenius $n$-exact categories have $(n+2)$-angulated structures.

Abstract

We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius $n$-exact category has a natural $(n+2)$-angulated structure in the sense of Gei\ss-Keller-Oppermann. We give several examples of $n$-abelian and $n$-exact categories which have appeared in representation theory, commutative ring theory, commutative and non-commutative algebraic geometry.