$n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories

TL;DR

This paper introduces $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories, axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories, and explores their structures and relationships within abelian and exact categories.

Contribution

It formalizes $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories and establishes their connection to $n\mathbb{Z}$-cluster tilting subcategories, providing a new framework in category theory.

Findings

Every $n\mathbb{Z}$-cluster tilting subcategory of an abelian or exact category has an $n\mathbb{Z}$-abelian or $n\mathbb{Z}$-exact structure.

Every small $n\mathbb{Z}$-abelian category arises from this construction.

Discussion on the characterization of $n\mathbb{Z}$-exact categories.

Abstract

In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this categories and show that every $n\mathbb{Z}$-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an $n\mathbb{Z}$-abelian (resp., $n\mathbb{Z}$-exact) category. Also we show that every small $n\mathbb{Z}$-abelian category arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.