Axiomatizing Subcategories of Abelian Categories

TL;DR

This paper develops intrinsic axioms to characterize important subcategories of abelian categories, including cluster tilting subcategories, and shows that any d-abelian category can be realized as a d-cluster tilting subcategory.

Contribution

It introduces axioms for subcategories like generating, cogenerating, and cluster tilting, and proves the equivalence of d-abelian categories to d-cluster tilting subcategories.

Findings

Intrinsic axioms characterize key subcategories.

Any d-abelian category is equivalent to a d-cluster tilting subcategory.

No projective generation assumption needed.

Abstract

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.