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<h1>XModAlg</h1>


<h2>Crossed Modules and Cat1-Algebras</h2>

<p>
    1.32</p>

<p>
    11 April 2025
  </p>

</div>
<p><b>
    Zekeriya Arvasi



  </b>
<br />Email: <span class="URL"><a href="mailto:zarvasi@ogu.edu.tr">zarvasi@ogu.edu.tr</a></span>
<br />Address: <br />Prof. Dr. Z. Arvasi <br /> Osmangazi University <br /> Arts and Sciences Faculty <br /> Department of Mathematics and Computer Science <br /> Eskisehir <br /> Turkey<br />
</p><p><b>
    Alper Odabas



  </b>
<br />Email: <span class="URL"><a href="mailto:aodabas@ogu.edu.tr">aodabas@ogu.edu.tr</a></span>
<br />Address: <br />Dr. A. Odabas <br /> Osmangazi University <br /> Arts and Sciences Faculty <br /> Department of Mathematics and Computer Science <br /> Eskisehir <br /> Turkey<br />
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">XModAlg</strong> package provides functions for computation with crossed modules of commutative algebras and cat<span class="SimpleMath">^1</span>-algebras.</p>

<p>Bug reports, suggestions and comments are, of course, welcome. Please submit an issue on GitHub at <span class="URL"><a href="https://github.com/gap-packages/xmodalg/issues/">https://github.com/gap-packages/xmodalg/issues/</a></span> or contact the second author at <span class="URL"><a href="mailto:aodabas@ogu.edu.tr">aodabas@ogu.edu.tr</a></span>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2014-2025, Zekeriya Arvasi and Alper Odabas.</p>

<p>The <strong class="pkg">XModAlg</strong> package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>

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<h3>Acknowledgements</h3>
<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> <a href="chapBib.html#biBGAPDoc">[LN17]</a> and <strong class="pkg">AutoDoc</strong> <a href="chapBib.html#biBAutoDoc">[GH17]</a> packages.</p>

<p>The procedure used to produce new releases uses the package <strong class="pkg">GitHubPagesForGAP</strong> <a href="chapBib.html#biBGitHubPagesForGAP">[Hor17]</a> and the package <strong class="pkg">ReleaseTools</strong>.</p>

<p>Both authors are very grateful to Chris Wensley (<span class="URL"><a href="https://github.com/cdwensley">https://github.com/cdwensley</a></span>) for helpful suggestions.</p>

<p>This work was partially supported by TÜBİTAK (The Scientific and Technical Research Council of Turkey), project number 107T542.</p>

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

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
</div>
<div class="ContChap"><a href="chap2.html#X85E897DA7DEFA8FB">2 <span class="Heading">Algebras and their Actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8313C8E07E64957A">2.1 <span class="Heading">Multipliers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8695ED0A7B9D2D3E">2.1-1 RegularAlgebraMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79D9464285963A52">2.1-2 IsAlgebraMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X874D72C17FDC0F57">2.1-3 MultiplierAlgebraOfIdealBySubalgebra </a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D3640FB8658B360">2.1-4 MultiplierAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FE8408D7F13C3B9">2.1-5 MultiplierHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7DD27C4F832785C0">2.2 <span class="Heading">Commutative actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87F6A0237B15AEC8">2.2-1 AlgebraAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79F1962A7F2230D3">2.2-2 AlgebraActionByMultipliers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7EAF09677CAE12D5">2.2-3 AlgebraActionBySurjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8530E1B27BC2FBB7">2.2-4 AlgebraActionByHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X854D23487A8FF78A">2.3 <span class="Heading">Algebra modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83A1091782FF581C">2.3-1 ModuleAsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X837793B87FFBA954">2.3-2 IsModuleAsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8235F36D7EB63BA3">2.3-3 ModuleToAlgebraIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C1C8B987B2167B9">2.3-4 AlgebraActionByModule</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X82E784EB7EBAEEA4">2.4 <span class="Heading">Actions on direct sums of algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86FC4981819374E4">2.4-1 DirectSumOfAlgebrasWithInfo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X800F0CFE7F431D92">2.4-2 Embedding</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86CCCCB4786AC96B">2.5 <span class="Heading">Other operations on algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A9527E97EC79B4F">2.5-1 SemidirectProductOfAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8536D21A80AFE08E">2.5-2 SemidirectProductOfAlgebrasInfo</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7960904E7A0536A8">2.6 <span class="Heading">Lists of algebra homomorphisms</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81C710788550185A">2.6-1 AllAlgebraHomomorphisms</a></span>
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</div>
<div class="ContChap"><a href="chap3.html#X85527BA8786CB7FC">3 <span class="Heading">Cat1-algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X811B6B0F8203F972">3.1 <span class="Heading">Definitions and examples</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B761CD9812972F6">3.1-1 Cat1Algebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F9561168414C58F">3.1-2 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82EC94BA7E7F8DEA">3.1-3 Cat1AlgebraSelect</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86E99B197E920C21">3.1-4 SubCat1Algebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DA775CA8296A7D8">3.2 <span class="Heading">Cat<span class="SimpleMath">^1-</span>algebra morphisms</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X860E29147DA143B5">3.2-1 Cat1AlgebraMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A218A0C7DBA8B63">3.2-2 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78AE603C857E4EBD">3.2-3 ImagesSource2DimensionalMapping</a></span>
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</div>
<div class="ContChap"><a href="chap4.html#X808C6B357F8BADC1">4 <span class="Heading">Crossed modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BB9D67179296AA0">4.1 <span class="Heading">Definition and Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X813D94F97D8E71A8">4.1-1 XModAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B31475D7C030075">4.1-2 XModAlgebraByIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CD49D5A84FB05FC">4.1-3 AugmentationXMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83C5D98B828DD1DD">4.1-4 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X814C89667B997FE8">4.1-5 XModAlgebraByMultiplierAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X824D4DD88323EF70">4.1-6 XModAlgebraBySurjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AB09A158148290F">4.1-7 XModAlgebraByBoundaryAndAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78400B837A2C8FB9">4.1-8 XModAlgebraByModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CA213397B334CBC">4.1-9 SubXModAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X866610BC7C41E1EC">4.2 <span class="Heading">(Pre-)Crossed Module Morphisms</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D575D07810640EB">4.2-1 XModAlgebraMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83F45A3E8554ABE9">4.2-2 Kernel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87F4D35A826599C6">4.2-3 Image</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B7975DA7E870690">4.2-4 SourceHom</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X7D65751085F46462">5 <span class="Heading">Conversion between cat1-algebras and crossed modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8617844F86989A78">5.1 <span class="Heading">Equivalent Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A7328237B8ABDD1">5.1-1 Cat1AlgebraOfXModAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EDFD11181CE143B">5.1-2 XModAlgebraOfCat1Algebra</a></span>
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</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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