Classifying substructures of extriangulated categories via Serre subcategories

Published in Appl. Categ. Structures, April 2021. Links: journal, arXiv:2005.13381

Citation: H. Enomoto, Classifying substructures of extriangulated categories via Serre subcategories, Appl. Categ. Structures 29 (2021), no. 6, 1005--1018.


In the previous exact-structure paper, I gave a classification of exact structures of a given additive category using defects in functor category. In this paper, we give analogous results for extriangulated categories introduced by Nakaoka-Palu.

However, since we cannot recover triangulated structures from the category of defects (they always coincide with the category of finitely presented functors), we cannot expect a full classification of any (ex)triangulated structures. Instead, I investigate substructures of a given extriangulated structure.

For a given extriangulated category, I classify all possible substructures on it. More precisely, substructures are in bijection with Serre subcategories of the category of defects. As an application to exact categories, I proved that the lattice of exact structures on a given additive category is isomorphic to the lattice of Serre subcategories of some abelian category.