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<h1>LinearAlgebraForCAP</h1>


<h2>Category of Matrices over a Field for CAP</h2>

<p>
    2025.09-01</p>

<p>
    17 September 2025
  </p>

</div>
<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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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X79A2BF518637FFB7">1 <span class="Heading">Category of Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X86EC0F0A78ECBC10">1.1 <span class="Heading">Constructors</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X79D9727280354457">1.1-1 MatrixCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C6DB0457A656406">1.1-2 VectorSpaceMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7EB162887B8F8708">1.1-3 VectorSpaceObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X818AE01E7C9D21A5">1.1-4 MatrixCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7A0CAF7F7CE754E7">1.1-5 MatrixCategory_as_CategoryOfRows</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C701DBF7BAE649A">1.2 <span class="Heading">Attributes</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7A25998E79499AE8">1.2-1 UnderlyingFieldForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8513433F7DA84E66">1.2-2 UnderlyingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X83D0E5BD8763C8FD">1.2-3 UnderlyingFieldForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X81C6A5F382A1D4F5">1.2-4 Dimension</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7D03633A7D98026B">1.3 <span class="Heading">GAP Categories</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7DAEE1D67CDFA35A">1.3-1 IsVectorSpaceMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7EF7321D794F9050">1.3-2 IsVectorSpaceObject</a></span>
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<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#X8104A77D7B5CCD4F">2.1 <span class="Heading">Basic Commands</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78D1062D78BE08C1">2.2 <span class="Heading">Functors</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X85B1F5077BE75340">2.3 <span class="Heading">Solving (Homogeneous) Linear Systems</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X852F59737EC57965">2.4 <span class="Heading">Homology object</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7CDB7FF27D734D2D">2.5 <span class="Heading">Liftable</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X797080CE873BB56D">2.6 <span class="Heading">Monoidal structure</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8218390A785F1EDB">2.7 <span class="Heading">MorphismFromSourceToPushout and MorphismFromFiberProductToSink</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X84410C637B3C1D0E">2.8 <span class="Heading">Opposite category</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7F34E29281DF3FCE">2.9 <span class="Heading">PreComposeList and PostComposeList</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X87F5609A7B33C58C">2.10 <span class="Heading">Split epi summand</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7DCD99628504B810">2.11 <span class="Heading">Kernel</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7DE20941803BFBD9">2.12 <span class="Heading">FiberProduct</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8245BF297DF9A3E7">2.13 <span class="Heading">WrapperCategory</span></a>
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