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<p>Given an extriangulated category <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\mathcal {C},\mathbb {E},\mathfrak {s})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo>,</mo> <mi mathvariant="double-struck">E</mi> <mo>,</mo> <mi mathvariant="fraktur">s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we introduce the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((3 \times 3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-lemma property for subfunctors of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">E</mi> </math></EquationSource> </InlineEquation> and prove that an additive subfunctor <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathbb {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathbb {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">E</mi> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\xi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月16日

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  • Juan C. Cala,
  • Shaira R. Hernández

摘要

Given an extriangulated category \((\mathcal {C},\mathbb {E},\mathfrak {s})\) ( C , E , s ) , we introduce the \((3 \times 3)\) ( 3 × 3 ) -lemma property for subfunctors of \(\mathbb {E}\) E and prove that an additive subfunctor \(\mathbb {F}\) F of \(\mathbb {E}\) 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 \(\xi \) ξ in \(\mathcal {C}\) C .