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<p>The problem of factorization of groups and monoids in a Cartesian monoidal category <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> is investigated. Two necessary and sufficient conditions are given in terms of so-called descent 1-cocycles for a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-monoid to be factorized via two <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-submonoids. In addition, a criterion is obtained for a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-monoid to be a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-group, from which the characterization of cocommutative <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-bimonoids that are Hopf <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathscr {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>-monoids is derived.</p>

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ON FACTORIZATION OF MONOIDS IN CARTESIAN CATEGORIES

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月1日

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  • Tamar Mesablishvili

摘要

The problem of factorization of groups and monoids in a Cartesian monoidal category \(\mathscr {C}\) C is investigated. Two necessary and sufficient conditions are given in terms of so-called descent 1-cocycles for a \(\mathscr {C}\) C -monoid to be factorized via two \(\mathscr {C}\) C -submonoids. In addition, a criterion is obtained for a \(\mathscr {C}\) C -monoid to be a \(\mathscr {C}\) C -group, from which the characterization of cocommutative \(\mathscr {C}\) C -bimonoids that are Hopf \(\mathscr {C}\) C -monoids is derived.