Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories

Abstract

It was shown recently that the heart H of a twin cotorsion pair ((S, T ), (U, V)) on an extriangulated category is semi-abelian. We provide a sufficient condition for the heart to be integral and another for the heart to be quasi-abelian. This unifies and improves the corresponding results for exact and triangulated categories. Furthermore, if T = U, then we show that the Gabriel-Zisman localisation of H at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair (S, T ). This generalises and improves the known result for triangulated categories, thereby providing new insights in the exact setting.

Abelian categories from triangulated categories via Nakaoka-Palu's localization

“…In many fields of mathematics containing representation theory, it is basic to study how to construct related abelian categories from a given triangulated category C. There are many researches in this context, for example [BBD,KR,KZ,Nak1,AN,BM1,BM2,Bel,Nak2,Pal,LN,HS]. Our study has its origin in the following result in [KZ,Thm.…”

“…In this section, we give some comments on results closely related to Theorem 3.8 which can be easily deduced from [Nak2,HS] (see [Liu2] for exact case). Their method depends on a use of the twin cotorsion pair Q := ((S, T ), (U , V)) which is defined to be a pair of two cotorsion pairs with S ⊆ U .…”

“…In this case, it directly follows from [Nak2,Thm. 6.3] and [HS,Thm. 5.6] that there exists a commutative diagram which is a special case of Theorem 3.8(4) as below:…”

“…Their construction of preabelian categories was improved by many authers, e.g. [Bel,Nak2,Liu2,LN,HS]. On the other hand, before [BM1,BM2], the category mod End C (T ) op has already been realized as a subfactor of C in [IYo]: an equivalence…”