A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories

TL;DR

The paper characterizes closed subfunctors of $E$ in extriangulated categories via a $3×3$-lemma property, extending A. Buan's abelian-category result to the extriangulated setting. It provides an elementary proof that avoids the Freyd–Mitchell embedding theorem and shows that a subfunctor $F$ is closed iff it satisfies the $3×3$-lemma property. As a consequence, saturated proper classes $ξ$ of $E$-triangles correspond bijectively to closed subfunctors $E_ξ$, providing new equivalent descriptions of saturated classes and a unified view of relative theories in extriangulated categories.

Abstract

Given an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we introduce the $3 \times 3$-lemma property for subfunctors of $\mathbb{E}$ and prove that an additive subfunctor $\mathbb{F}$ of $\mathbb{E}$ is closed if, and only if, it satisfies this condition. This characterization extends a well known result by A. Buan (for abelian categories) to extriangulated categories. As an application of this result, we get a new equivalent condition to describe saturated proper classes $ξ$ in $\mathcal{C}$.

Theorems & Definitions (16)

• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• Lemma 2.1
• proof
• Definition 2.2
• Theorem 2.3
• proof
• Corollary 2.4