Phantom ideals and cotorsion pairs in extriangulated categories

TL;DR

This paper introduces relative phantom morphisms in extriangulated categories and establishes bijective correspondences between various classes of ideals and subfunctors, advancing the understanding of cotorsion pairs in this context.

Contribution

It defines and studies relative phantom morphisms in extriangulated categories and links precovering ideals with subfunctors, providing new structural insights.

Findings

Established bijections between special precovering and preenveloping ideals.

Connected additive subfunctors with enough special injective and projective morphisms.

Extended the framework of 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.