Higher-dimensional generalization of abelian categories via DG-categories

TL;DR

This paper introduces a higher-dimensional generalization of abelian categories within DG-categories, unifying classical and stable cases, and develops their structural properties and factorization theories.

Contribution

It defines abelian n-truncated DG-categories as a new higher-dimensional analogue of abelian categories, extending classical and stable cases and establishing their structural framework.

Findings

Homotopy categories of abelian n-truncated DG-categories are extriangulated and pretriangulated.

Develops a theory including epi-mono factorizations in these categories.

Unifies classical abelian categories and stable DG-categories under a higher-dimensional framework.

Abstract

In this paper, we introduce abelian $n$-truncated DG-categories as an $n$-dimensional analogue of abelian categories in the setting of DG-categories. When $n=1$, this recovers ordinary abelian categories, and when $n=\infty$, it corresponds to stable DG-categories. This notion serves as a DG-categorical analogue of abelian $(n,1)$-categories in the context of $(n,1)$-categories. We show that the homotopy categories of abelian $n$-truncated DG-categories acquire the structure of extriangulated and pretriangulated categories. Furthermore, we develop a general theory of abelian $n$-truncated DG-categories, including the analogues of the existence epi-mono factorizations of morphisms, as in classical abelian categories.