Higher exact dg-categories

TL;DR

This paper introduces the concept of n-exact dg-categories, extending Chen's exact dg-categories to higher dimensions, and explores their properties and connections to n-cluster tilting subcategories.

Contribution

It defines n-exact dg-categories, shows their homotopy categories admit n-exangulated structures under certain conditions, and relates them to n-cluster tilting subcategories.

Findings

Homotopy categories of n-exact dg-categories admit n-exangulated structures.

n-cluster tilting subcategories naturally carry n-exact dg-structures.

Framework generalizes n-exangulated categories without vanishing conditions.

Abstract

We introduce the notion of an $n$-exact dg-category. This notion provides a higher analogue of Chen's exact dg-category, in the sense that the case where $n$ equals 1 recovers exact dg-categories. We prove that, under a suitable vanishing condition on the cohomologies of $\mathrm{Hom}$-complexes of an $n$-exact dg-category $\mathscr{A}$, its homotopy category admits a natural $n$-exangulated structure. Thus $n$-exact dg-categories provide dg-enhancements of $n$-exangulated categories. At the same time, our framework can be regarded as a dg-categorical generalization of $n$-exangulated categories applicable even without the vanishing condition. In the latter part of the article, we show that an $n$-cluster tilting subcategory of an exact dg-category naturally carries the structure of an $n$-exact dg-category. This result indicates that $n$-exact dg-structures provide an intrinsic dg-categorical axiomatization of $n$-cluster tilting subcategories, highlighting the advantages of studying dg-generalizations of $n$-exangulated categories.