Ideal approximation in $n$-exangulated categories

TL;DR

This paper explores ideal approximation theory within $n$-exangulated categories, introducing new concepts like $n$-ideal cotorsion pairs and $n$-$$-phantom morphisms, and proves Salce's Lemma in certain subcategories.

Contribution

It introduces and studies $n$-ideal cotorsion pairs and $n$-$$-phantom morphisms in $n$-exangulated categories, extending approximation theory.

Findings

Introduction of $n$-ideal cotorsion pairs and $n$-$$-phantom morphisms.

Proved Salce's Lemma in a nicely embedded $n$-cluster tilting subcategory.

Established properties of ideal approximation in $n$-exangulated categories.

Abstract

In this paper, we study the ideal approximation theory associated to almost $n$-exact structures in the $n$-exangulated category. The notions of $n$-ideal cotorsion pairs and $n$-$\mathbb{F}$-phantom morphisms are introduced and studied. In particular, let $\mathscr{C}$ be an extriangulated category which satisfies the condition (WIC) and $\mathcal{T}$ be a nicely embedded $n$-cluster tilting subcategory of $\mathscr{C}$, we prove Salce's Lemma in $\mathcal{T}$.