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<h1>ModulePresentationsForCAP</h1>


<h2>Category R-pres for CAP</h2>

<p>
    2025.09-01</p>

<p>
    22 September 2025
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<p><b>
    Sebastian Gutsche




  </b>
<br />Email: <span class="URL"><a href="mailto:sebastian.gutsche@gmail.com">sebastian.gutsche@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="https://sebasguts.github.io/">https://sebasguts.github.io/</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
    Sebastian Posur




  </b>
<br />Email: <span class="URL"><a href="mailto:sebastian.posur@uni-muenster.de">sebastian.posur@uni-muenster.de</a></span>
<br />Homepage: <span class="URL"><a href="https://sebastianpos.github.io">https://sebastianpos.github.io</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
    Fabian Zickgraf




  </b>
<br />Email: <span class="URL"><a href="mailto:f.zickgraf@dashdos.com">f.zickgraf@dashdos.com</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/zickgraf/">https://github.com/zickgraf/</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57068 Siegen<br /> Germany<br />
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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7B8C95CA7DA733B4">1 <span class="Heading">Module Presentations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X78D1062D78BE08C1">1.1 <span class="Heading">Functors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F7AC44478418555">1.1-1 FunctorStandardModuleLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87AEC1177DB7F50D">1.1-2 FunctorStandardModuleRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8427C0B17A445822">1.1-3 FunctorGetRidOfZeroGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F1E779D8003146B">1.1-4 FunctorGetRidOfZeroGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X819A04517B3601C0">1.1-5 FunctorLessGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F80AE9B7EC07198">1.1-6 FunctorLessGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X877B7ACE87E1BEC2">1.1-7 FunctorDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D56611D7BF91B54">1.1-8 FunctorDualRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C9901F8851FD24A">1.1-9 FunctorDoubleDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8016306881444DCA">1.1-10 FunctorDoubleDualRight</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7D03633A7D98026B">1.2 <span class="Heading">GAP Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X79DBCB747E91FB70">1.2-1 IsLeftOrRightPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85E26CFF86855B6B">1.2-2 IsLeftPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X873EFE29849F6998">1.2-3 IsRightPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BB95B7A7EB96854">1.2-4 IsLeftOrRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C71B8D17C60C6B5">1.2-5 IsLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7DBF478D7EE3FE63">1.2-6 IsRightPresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X86EC0F0A78ECBC10">1.3 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87010AB6819736C8">1.3-1 PresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C9B36AD7B9CCC8D">1.3-2 AsMorphismBetweenFreeLeftPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F1AD55C852BE617">1.3-3 AsMorphismBetweenFreeRightPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BE01A1381744627">1.3-4 AsLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X780443B07F43AA1C">1.3-5 AsRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F345A2A87ABE417">1.3-6 FreeLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85536E4E85D15252">1.3-7 FreeRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X86F926B27C579E66">1.3-8 UnderlyingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D8E30E486A08439">1.3-9 UnderlyingHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B5539AB8541F618">1.3-10 Annihilator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87946F997AD1005A">1.3-11 LeftPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BDF988F7FFEAB8C">1.3-12 RightPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8145816A85BA2680">1.3-13 LeftPresentations_as_FreydCategory_CategoryOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7ED9DFEB82DE9653">1.3-14 RightPresentations_as_FreydCategory_CategoryOfColumns</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C701DBF7BAE649A">1.4 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8157F3E8847B15E1">1.4-1 UnderlyingHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X83CA6F06832162B7">1.4-2 UnderlyingMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X81CDBC6E7DBB4EA0">1.5 <span class="Heading">Non-Categorical Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8508310C7E908093">1.5-1 StandardGeneratorMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7EF1493A7D341F5E">1.5-2 CoverByFreeModule</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X836749D8814FEEE6">1.6 <span class="Heading">Natural Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85D3DB1F856D05EF">1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B64AF718133C945">1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7CA2B84E7F933125">1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7A2DCBD6844093E3">1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B331B0A86010185">1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X834AC0FD825FCD2F">1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E37CB058378CBEE">1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F92B4448041A68C">1.6-8 NaturalTransformationFromIdentityToDoubleDualRight</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X7967FE8E7BBDF485">2 <span class="Heading">Examples and Tests</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7FFB58F67850769C">2.1 <span class="Heading">Annihilator</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X86E8A9537A87B4EC">2.2 <span class="Heading">Intersection of Submodules</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8296A25779A52244">2.3 <span class="Heading">Koszul Complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X808DC49C7ED99B52">2.4 <span class="Heading">Monoidal Categories</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X828DF9B1782D1AF6">2.5 <span class="Heading">Closed Monoidal Structure</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7C3B1694803034A2">2.6 <span class="Heading">Projectivity test</span></a>
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