What does this research mean for the field?

An additive subfunctor of an extriangulated category is closed if and only if it satisfies the (3×3)-lemma property. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to characterize closed subfunctors using the 3x3-lemma property in extriangulated categories.

February 19, 2026Open Access

A characterization of closed subfunctors through (3 3) -lemma property in extriangulated categories

Key Points

The aim is to characterize closed subfunctors using the 3x3-lemma property in extriangulated categories.
Introduced the 3x3-lemma property for subfunctors of the extriangulated category (C, E, s).
Proved that an additive subfunctor F of E is closed if it satisfies the 3x3-lemma property.
Extended existing results from abelian categories to extriangulated categories.
Confirmed the conditions under which an additive subfunctor is closed.
Derived a new equivalent condition for describing saturated proper classes in C.

Abstract

Abstract Given an extriangulated category (C, E, s) (C, E, s), we introduce the (3 3) (3 × 3) -lemma property for subfunctors of E E and prove that an additive subfunctor F F of E 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 C C.

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