Localization of extriangulated categories

Abstract

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories by means of Hovey twin cotorsion pairs, and the localization of exact categories by two-sided admissibly percolating subcategories.

s-Recollements and its localization

“…Theorem 3.1. [25] Let (A , E, s) be an extriangulated category and I ⊆ M A a set of morphisms which satisfies (M0), (MR1), (MR2), (MR3) and (MR4). Then the localization of A by I gives an extriangulated category ( A , E, s) equipped with an exact functor…”

“…At last, we will remark that if (A , B, C ) is an s-recollement, then C is a localization of B in the sense of [25,Theorem 3.5].…”

“…Let I = j * −1 (IsoC ). By Corollary [25,Corollary 3.8], there is a unique exact functor j * : B → C such that j * = j * Q, where Q is a composition of canonical quotient functor p : B → B/[N I ] and localization functor Q : B/[N I ] → B, see [25] for the detail. It is clearly that…”

“…Proposition 5.5. [25] Let A be an extriangulated category satisfying (WIC) condition, or A be an exact category. If N is a biresolving thick subcategory of A , then the following hold.…”

“…[25] Let (A , E, s) be an extriangulated category (1) A full additive subcategory N ⊆ A is called a thick subcategory if it satisfies the following conditions. (a) N ⊆ A is closed by isomorphisms and direct summands.…”