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<h1>groupoids</h1>
<h2>Calculations with finite groupoids and their homomorphisms</h2>
<p>
1.79</p>
<p>
11 September 2025
</p>
</div>
<p><b>
Emma J. Moore
</b>
</p><p><b>
Chris Wensley
</b>
<br />Email: <span class="URL"><a href="mailto:cdwensley.maths@btinternet.com">cdwensley.maths@btinternet.com</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/cdwensley">https://github.com/cdwensley</a></span>
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">groupoids</strong> package provides functions for computation with groupoids (categories with every arrow invertible) and their morphisms; for graphs of groups, and graphs of groupoids. The most basic structure introduced is that of <em>magma with objects</em>; followed by <em>semigroup with objects</em>; then <em>monoid with objects</em>; and finally <em>groupoid</em>, which is a <em>group with objects</em>.</p>
<p>It provides normal forms for Free Products with Amalgamation and for HNN-extensions when the initial groups have rewrite systems and the subgroups have finite index.</p>
<p>The <strong class="pkg">groupoids</strong> package was originally implemented in 2000 (as <strong class="pkg">GraphGpd</strong>) when the first author was studying for a Ph.D. in Bangor.</p>
<p>The package was then renamed <strong class="pkg">Gpd</strong> and version 1.07 was released in July 2011, ready for <strong class="pkg">GAP</strong> 4.5.</p>
<p><strong class="pkg">Gpd</strong> became an accepted <strong class="pkg">GAP</strong> package in May 2015.</p>
<p>In April 2017 the package was renamed again, as <strong class="pkg">groupoids</strong>.</p>
<p>Later versions implement many of the constructions described in the paper <a href="chapBib_mj.html#biBAlWe">[AW10]</a> for automorphisms of groupoids.</p>
<p>Bug reports, comments, suggestions for additional features, and offers to implement some of these, will all be very welcome.</p>
<p>Please submit any issues at <span class="URL"><a href="https://github.com/gap-packages/groupoids/issues/">https://github.com/gap-packages/groupoids/issues/</a></span> or send an email to the second author at <span class="URL"><a href="mailto:cdwensley.maths@btinternet.com">cdwensley.maths@btinternet.com</a></span>.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2000-2025, Emma Moore and Chris Wensley.</p>
<p>The <strong class="pkg">groupoids</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>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This documentation was prepared using the <strong class="pkg">GAPDoc</strong> <a href="chapBib_mj.html#biBGAPDoc">[LN17]</a> and <strong class="pkg">AutoDoc</strong> <a href="chapBib_mj.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_mj.html#biBGitHubPagesForGAP">[Hor17]</a> and the package <strong class="pkg">ReleaseTools</strong>.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X8361CEA4856430C6">2 <span class="Heading">Many-object structures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7920BF45828AE5DE">2.1 <span class="Heading">Magmas with objects; arrows</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7C51D7847BD23284">2.1-1 MagmaWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X87D1C0BC86749E6D">2.1-2 IsDomainWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X86247DFC8242CEBE">2.1-3 Arrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7B3352148103FD05">2.1-4 IsSinglePieceDomain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7FDF94E8791B9BC1">2.2 <span class="Heading">Semigroups with objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X8519787D833653A7">2.2-1 SemigroupWithObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X86C32E6981557ED0">2.3 <span class="Heading">Monoids with objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7A271B1A7F2FF245">2.3-1 MonoidWithObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X842D1B0C7A76CC15">2.4 <span class="Heading">Generators of magmas with objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X83239B307EE95054">2.4-1 GeneratorsOfMagmaWithObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7C7007207FD91BB0">2.5 <span class="Heading">Structures with more than one piece</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X781FB06382E08115">2.5-1 UnionOfPieces</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X78FC7902804CED8E">3 <span class="Heading">Mappings of many-object structures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X82F856A086B93832">3.1 <span class="Heading">Homomorphisms of magmas with objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X86E00FEA7FF38FEA">3.1-1 MagmaWithObjectsHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C4D1AEE80D41A35">3.2 <span class="Heading">Homomorphisms of semigroups and monoids with objects</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X795C8DE37AED7B44">3.3 <span class="Heading">Homomorphisms to more than one piece</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7AE44D9485EB50F1">3.3-1 HomomorphismByUnion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7C053B0379DDCE13">3.3-2 IsInjectiveOnObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B9D71BB7BAEFAAA">3.4 <span class="Heading">Mappings defined by a function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81B64F9A855D265F">3.4-1 MappingWithObjectsByFunction</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X82F6A1AB798648F4">4 <span class="Heading">Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7AC321DC7E3C6167">4.1 <span class="Heading">Groupoids: their properties and attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8406913B7ED86CFE">4.1-1 SinglePieceGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X813B084184AAA890">4.1-2 ObjectList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8511D3EE845CC930">4.1-3 IsPermGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8681D15C7CA15F87">4.1-4 UnionOfPieces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X855F318181808814">4.1-5 HomogeneousGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78E8FE0C7F599F5A">4.1-6 DirectProductOp</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7F23BE3F85C9BA06">4.2 <span class="Heading">Groupoid elements; stars; costars; homsets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D028B3B8385ED07">4.2-1 GroupoidElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78C3228682279032">4.2-2 IdentityArrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84F59A2687C62763">4.2-3 Order</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B561BAE7D471C60">4.2-4 ObjectStar</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7BDBA72C852C4625">4.3 <span class="Heading">Subgroupoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80780D287EDDB7BB">4.3-1 Subgroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8600FA7B8051C9AC">4.3-2 SubgroupoidByObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X873E0914871858E8">4.3-3 SubgroupoidWithRays</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AEA9BF780CD6957">4.3-4 SubgroupoidByPieces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78C7A83980BAE937">4.3-5 PiecePositions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7FE8F09A7C8CEC5B">4.3-6 FullTrivialSubgroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7ACFBE7D855A82A3">4.3-7 DiscreteSubgroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A9F29BD7D64DCC1">4.3-8 SinglePieceSubgroupoidByGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X831AA9E8780235F2">4.4 <span class="Heading">Left, right and double cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8412ABD57986B9FC">4.4-1 RightCoset</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8653FC9786E3209A">4.5 <span class="Heading">Conjugation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D21FB1A7D21FB1A"><code>4.5-1 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78218F357BFEDE82">4.5-2 ConjugateGroupoid</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X86DD749283068705">4.6 <span class="Heading">Groupoids formed using isomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B066DE28574A768">4.6-1 GroupoidByIsomorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X795DEB3D825B2120">4.7 <span class="Heading">Groupoids whose objects form a monoid</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X805FA9F687C7F2A7">4.7-1 SinglePieceGroupoidWithRays</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E399664868937C8">4.7-2 RightActionGroupoid</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X879ED2A7878B5A7A">5 <span class="Heading">Homomorphisms of Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85D28690823636CD">5.1 <span class="Heading">Homomorphisms from a connected groupoid</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X870288C6803D9005">5.1-1 GroupoidHomomorphismFromSinglePiece</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7BFF46FD8308846F">5.2 <span class="Heading">Properties and attributes of groupoid homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X83DDA9528396451A">5.2-1 <span class="Heading">Properties of a groupoid homomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X870565948381B921">5.2-2 <span class="Heading">Attributes of a groupoid homomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X85A99D8A83511DDE">5.2-3 RootGroupHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X837EEB9578EA1A78">5.2-4 ImagesOfObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X791DC3F3852A6422">5.2-5 ImageElementsOfRays</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8391526C7A02FF19">5.2-6 MappingToSinglePieceData</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7C6775A178DB9D39">5.2-7 ObjectGroupHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7DBC80D8817CDA04">5.3 <span class="Heading">Special types of groupoid homomorphism</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X855DB6C8815CDCC3">5.3-1 InclusionMappingGroupoids</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7946BC207D455A98">5.3-2 RestrictedMappingGroupoids</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X83B8AB6D7C9DB264">5.3-3 IsomorphismNewObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87C54BB581181400">5.3-4 IsomorphismStandardGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7CF33CCD8278A904">5.3-5 IsomorphismPermGroupoid</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7EAAB2A97E097FB6">5.4 <span class="Heading">Homomorphisms to a connected groupoid</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8700B5977B2513C1">5.4-1 HomomorphismToSinglePiece</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87225D86785DEBC2">5.4-2 GroupoidHomomorphismFromHomogeneousDiscrete</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X795C8DE37AED7B44">5.5 <span class="Heading">Homomorphisms to more than one piece</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X847094F478113EA6">5.5-1 HomomorphismByUnion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7E2C6BCC7AAB25BE">5.5-2 IsomorphismGroupoids</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X803E01577A2B37D2">6 <span class="Heading">Automorphisms of Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7935832881320667">6.1 <span class="Heading">Automorphisms of single piece groupoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79DA704A8051CB72">6.1-1 GroupoidAutomorphismByObjectPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X86BD61567B41B139">6.1-2 GroupoidInnerAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8278B93E7F78038C">6.1-3 <span class="Heading">Automorphisms of a groupoid with rays</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7DB9D5737EB8C260">6.1-4 AutomorphismGroupOfGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7EC237ED7E1978B0">6.1-5 <span class="Heading">Inner automorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X80F9594481CA9FDE">6.1-6 GroupoidAutomorphismByGroupAutos</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A93C6507BC697CD">6.1-7 AutomorphismGroupoidOfGroupoid</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X8003AF117B956D16">6.2 <span class="Heading">Matrix representations of groupoids</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7DB8E7EF7A51F1BE">6.3 <span class="Heading">Groupoid actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C4DCB287B3DD0CD">6.3-1 GroupoidActionByConjugation</a></span>
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</div>
<div class="ContChap"><a href="chap7_mj.html#X78063DC8847554B4">7 <span class="Heading">Graphs of Groups and Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7D554C5D7FDC3D02">7.1 <span class="Heading">Digraphs</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85BD6D2584D8A22F">7.1-1 FpWeightedDigraph</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7BAFCA3680E478AE">7.2 <span class="Heading">Graphs of Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8130246E854BC5D9">7.2-1 GraphOfGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X847464677F641527">7.2-2 IsGraphOfFpGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7B036C2B84E48BB1">7.2-3 RightTransversalsOfGraphOfGroups</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7BD9DCF87FB3E0AF">7.3 <span class="Heading">Words in a Graph of Groups and their normal forms</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X87937AE47C5B1018">7.3-1 GraphOfGroupsWord</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78FA7C3F831AA6E4">7.3-2 ReducedGraphOfGroupsWord</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7D99A7B37B36BAA8">7.4 <span class="Heading">Free products with amalgamation and HNN extensions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X795AB71F7E370119">7.4-1 FreeProductWithAmalgamation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86F5F9787E3644FD">7.4-2 ReducedImageElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7CB0F120804A8DED">7.4-3 HnnExtension</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X78108FB4814AE887">7.5 <span class="Heading">GraphsOfGroupoids and their Words</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8739267678808E85">7.5-1 GraphOfGroupoids</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7EC1A9067FC91255">7.5-2 GraphOfGroupoidsWord</a></span>
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<div class="ContChap"><a href="chap8_mj.html#X83B7E8A287C9284A">8 <span class="Heading">Double Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X780DA94780938851">8.1 <span class="Heading">Single piece double groupoids</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F5D722C83CB9139">8.1-1 SinglePieceBasicDoubleGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X823A3A7481B90EB7">8.1-2 SquareOfArrows</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X83DF8EEE7BA72AA3">8.1-3 IsCommutingSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F99A468819D7759">8.1-4 TransposedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D3737FA7E9E3ECA">8.1-5 HorizontalProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X873F01287A2DC41F">8.1-6 VerticalProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X79E6E4E97A23C257">8.1-7 HorizontalIdentities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7B8DEF5A7B5AEC46">8.1-8 <span class="Heading">Horizontal and vertical groupoids in <span class="SimpleMath">\(\Box(G)\)</span></span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7F319AA17ED13024">8.2 <span class="Heading">Double groupoids with more than one piece</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8589EFAE7935BEA6">8.2-1 UnionOfPieces</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X78E24657817C4D30">8.3 <span class="Heading">Generators of a double groupoid</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X80009143808725D5">8.3-1 DoubleGroupoidWithTrivialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D135AFB83995A32">8.3-2 DoubleGroupoidWithSingleObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X85A6C1A685357E66">8.3-3 <span class="Heading">What is the double groupoid generated by a set of squares?</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X80E0318F78D2AD28">8.4 <span class="Heading">Starting with two groupoids</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X83F6AFC185148621">8.5 <span class="Heading">Double groupoid homomorphisms</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X87752436787B199D">8.5-1 DoubleGroupoidHomomorphism</a></span>
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<div class="ContChap"><a href="chap9_mj.html#X7F2BA6DC7AD84FBC">9 <span class="Heading">Technical Notes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7C5B725A7AED3C8D">9.1 <span class="Heading">Many object structures</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7A4760E887C9C968">9.2 <span class="Heading">Many object homomorphisms</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap10_mj.html#X810C43BC7F63C4B4">10 <span class="Heading">Development History</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X8192EA4C7B7CC5CD">10.1 <span class="Heading">Versions of the Package</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X82148FB77CC3CCC0">10.2 <span class="Heading">What needs to be done next?</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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