Recollements of extriangulated categories

Abstract

We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement (A, B, C) of extriangulated categories, we show that cotorsion pairs in A and C induce cotorsion pairs in B under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomenon when it is applied to abelian categories.

Gluing n-tilting and n-cotilting subcategories

“…We recall the definition of recollement of extriangulated categories from [WWZ]. We only state the settings that we need, for details, one can see [WWZ,Section 3].…”

“…Proposition 2.5. [WWZ,Proposition 3.4] Let (A, B, C) be a recollement of extriangulated categories and X ∈ B. Then the following statements hold.…”

Resolving subcategories and dimensions in recollements of extriangulated categories

“…(2) is a dual of (1). In extriangulated categories, the notions of the left (right) exact sequence (resp., functor) and compatible morphisms can be founded in [24] for details. Also the notion of extriangulated functor between two extriangulated categories can be found in [5].…”

“…Now we recall the concept of recollements of extriangulated categories [24], which gives a simultaneous generalization of recollements of triangulated categories and abelian categories (see [3,7]).…”

“…They are closely related each other, and possess similar properties in many aspects. In order to give a simultaneous generalization of recollements of abelian categories and triangulated categories, Wang, Wei and Zhang [24] introduced the notion of recollements of extriangulated categories.…”