An enhanced extriangulated subquotient

TL;DR

This paper develops a framework for enhancing extriangulated categories with dg structures, generalizing existing notions and enabling broad localization techniques including ideal and Verdier quotients.

Contribution

It introduces the concept of extriangulated subquotients and cohomological envelopes, unifying various quotient constructions within an enhanced dg categorical setting.

Findings

Compatible dg enhancements for extriangulated quotients

Introduction of extriangulated subquotients for localization

Generalization of ideal and Verdier quotients in dg framework

Abstract

Bondal-Kapranov's notion of enhanced triangulated categories behaves well in the framework of localization theory, in the sense that the Verdier quotient of triangulated categories can be lifted to the Drinfeld dg quotient of pretriangulated dg categories. In this paper, we develop a parallel enhancement for Nakaoka-Palu's notion of extriangulated categories, which unifies exact and triangulated categories. The enhancement of extriangulated categories was recently initiated by Xiaofa Chen under the name exact dg categories. Moreover, it is known that certain ideal quotients of extriangulated categories remain extriangulated, and that such ideal quotients admit dg enhancements via the dg quotient of the corresponding connective exact dg category. Motivated by Chen's framework of enhanced extriangulated categories, we introduce the concept of a cohomological envelope of an exact dg category and generalize his construction of the enhanced ideal quotient. We show that the dg quotient of exact dg categories, when passing to cohomological envelopes and their substructures -- referred to as exact dg subquotients -- is compatible with a broad class of extriangulated quotients in the sense of Nakaoka-Ogawa-Sakai. To further clarify the scope of our approach, we formulate the notion of an extriangulated subquotient, which enables the localization of any extriangulated category by extension-closed subcategories. This construction encompasses not only ideal and Verdier quotients, but also the quotient of an exact category by a biresolving subcategory. Notably, the extriangulated subquotient admits a natural lifting to the exact dg subquotient.