Extriangulated ideal quotients and Gabriel-Zisman localizations

TL;DR

This paper demonstrates that under certain conditions, Gabriel-Zisman localizations of extriangulated categories can be realized as ideal quotients, which are also extriangulated, preserving their structure and relating to known quotient constructions.

Contribution

It establishes conditions under which localizations are equivalent to ideal quotients within extriangulated categories, extending the understanding of their structure and relations to other quotient frameworks.

Findings

Localization as ideal quotient is extriangulated

Equivalence preserves extriangulated structure

Connections to Hovey twin cotorsion pairs and Verdier quotients

Abstract

Let $(\mathcal B,\mathbb{E},\mathfrak{s})$ be an extriangulated category and $\mathcal S$ be an extension closed subcategory of $\mathcal B$. In this article, we prove that the Gabriel-Zisman localization $\mathcal B/\mathcal S$ can be realized as an ideal quotient inside $\mathcal B$ when $\mathcal S$ satisfies some mild conditions. The ideal quotient is an extriangulated category. We show that the equivalence between the ideal quotient and the localization preserves the extriangulated category structure. We also discuss the relations of our results with Hovey twin cotorsion pairs and Verdier quotients.