Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

Abstract

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if (C , E, s) is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of C ; (2) special preenveloping ideals of C ; (3) additive subfunctors of E having enough special injective morphisms; and (4) additive subfunctors of E having enough special projective morphisms. Moreover, we show that if (C , E, s) is an extriangulated category with enough injective objects and projective morphisms, then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of C ; (2) all additive subfunctors of E having enough special injective objects.

Resolution Dimension Relative to Resolving Subcategories in Extriangulated Categories

“…As a similar argument to that of the diagram (15), we obtain that the E-triangles 18) are in ξ. Thus, (18) is the second desired E-triangle in ξ with Hres.dim W ≤ n and X ∈ X.…”

“…In [12], Nakaoka and Palu introduced the notion of extriangulated categories as a simultaneous generalization of exact categories and extension-closed subcategories of triangulated categories. After that, the study of extriangulated categories has become an active topic, and up to now, many results on exact categories and triangulated categories can be unified in the same framework, e.g., see [8,[12][13][14][15][16]. Recently, Hu, Zhang, Zhou [13] studied a relative homological algebra in an extriangulated category (C , E, s) which parallels the relative homological algebra in triangulated categories and exact categories.…”

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

“…Extriangulated categories, recently introduced in [72], axiomatize extension‐closed subcategories of triangulated categories in a (moderately) similar way that Quillen's exact categories axiomatize extension‐closed subcategories of abelian categories. They appear in representation theory in relation with cotorsion pairs [29, 58, 59, 104], with Auslander–Reiten theory [51], with cluster algebras, mutations, or cluster‐tilting theory [29, 63–65, 83, 106], with Cohen–Macaulay dg‐modules in the remarkable [53]. We also note the generalization, called $$n$$‐exangulated categories [47, 48], to a version suited for higher homological algebra.…”

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

“…Extriangulated categories (see Definition 2.17) were introduced in [25] as a simultaneous generalisation of exact and triangulated categories. Many results (see, for example, [16], [35], [15], [21], [22], [33], [34], [36], [37]) that hold for exact and triangulated categories have been unified, i.e. shown to hold for extriangulated categories in general.…”