One-sided Frobenius pairs in extriangulated categories

TL;DR

This paper introduces new concepts of Frobenius pairs, cotorsion pairs, and Auslander-Buchweitz contexts in extriangulated categories, establishing their relationships and generalizing previous results from triangulated and abelian categories.

Contribution

It defines and relates Frobenius pairs, cotorsion pairs, and Auslander-Buchweitz contexts in extriangulated categories, extending known theories to a broader categorical framework.

Findings

Constructed left cotorsion pairs from left n-cotorsion pairs.

Established a one-to-one correspondence between Frobenius pairs and Auslander-Buchweitz contexts.

Generalized existing results from triangulated and abelian categories.

Abstract

Let $\mathscr{C}$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $\xi$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also study the relation between a certain class of cotorsion pairs and that of $n$-cotorsion pairs. These work generalize Ma-Zhao-Huang's results in triangulated categories and partially generalize Becerril-Mendoza-P\'{e}rez-Santiago's results in abelian categories.