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<p>We prove that the Leibniz PROP is isomorphic as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10485_2025_9820_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Bbbk \)</EquationSource> </InlineEquation>-linear categories (not as monoidal categories) to the symmetric crossed presimplicial algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10485_2025_9820_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Bbbk [(\Delta ^+)^{op} \mathbb {S}]\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10485_2025_9820_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta ^+\)</EquationSource> </InlineEquation> is the skeletal category of finite well-ordered sets with surjections, but the distributive law between <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10485_2025_9820_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\((\Delta ^+)^{op}\)</EquationSource> </InlineEquation> and the symmetric groups <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10485_2025_9820_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {S} = \bigsqcup _{n\ge 1} S_n\)</EquationSource> </InlineEquation> is not the standard one.</p>

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The Leibniz PROP is a Crossed Presimplicial Algebra

  • 期刊论文

  • 发表日期：

    2025年7月30日

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  • Murat Can Aşkaroğulları,
  • Atabey Kaygun

摘要

We prove that the Leibniz PROP is isomorphic as \(\Bbbk \) -linear categories (not as monoidal categories) to the symmetric crossed presimplicial algebra \(\Bbbk [(\Delta ^+)^{op} \mathbb {S}]\) where \(\Delta ^+\) is the skeletal category of finite well-ordered sets with surjections, but the distributive law between \((\Delta ^+)^{op}\) and the symmetric groups \(\mathbb {S} = \bigsqcup _{n\ge 1} S_n\) is not the standard one.