Classifying substructures of extriangulated categories via Serre subcategories

TL;DR

This paper classifies substructures of extriangulated categories using Serre subcategories of defect categories, establishing a bijection and linking exact structures to Serre subcategories.

Contribution

It introduces a novel classification method for substructures of extriangulated categories via Serre subcategories of defect categories.

Findings

Substructures correspond bijectively to Serre subcategories.

Poset of exact structures is isomorphic to Serre subcategories.

Provides a new perspective on classifying additive category structures.

Abstract

We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abelian category.