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<h1>CAP</h1>


<h2>Categories, Algorithms, Programming</h2>

<p>
    2025.09-04</p>

<p>
    22 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>
    Øystein Skartsæterhagen




  </b>
<br />Email: <span class="URL"><a href="mailto:oysteini@math.ntnu.no">oysteini@math.ntnu.no</a></span>
<br />Homepage: <span class="URL"><a href="http://www.math.ntnu.no/~oysteini/">http://www.math.ntnu.no/~oysteini/</a></span>
<br />Address: <br />NTNU<br /> Institutt for matematiske fag<br /> 7491 Trondheim<br /> Norway<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.html#X7C99ABCB7DEEDE2A">1 <span class="Heading">CAP Categories</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7CC6903E78F24167">1.1 <span class="Heading">Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X80E47D0E86A9F916">1.1-1 IsCapCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7E2CFDC482566454">1.1-2 IsCapCategoryCell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DAEF54180094D1E">1.1-3 IsCapCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84413C4978765A3F">1.1-4 IsCapCategoryMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F3B964286D98710">1.1-5 IsCapCategoryTwoCell</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A853C6D8096E14B">1.2 <span class="Heading">Categorical properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7D6EA1307DFCCE46">1.2-1 AddCategoricalProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X806B382F8077EB7B">1.2-2 IsObjectFiniteCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8296B39C789BE6A9">1.2-3 IsFiniteCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8347FF9C81E7385C">1.2-4 IsFinite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8572BE547C73D308">1.2-5 IsEquivalentToFiniteCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X83F78D2C7AAC115D">1.2-6 IsEquippedWithHomomorphismStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X829DB5497D58C7E5">1.2-7 IsCategoryWithDecidableLifts</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C22FA787967E694">1.2-8 IsCategoryWithDecidableColifts</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C2B7A5F87EE5322">1.2-9 IsCategoryWithInitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X828327367FA16BDD">1.2-10 IsCategoryWithTerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BFEEC3779A5EE2F">1.2-11 IsCategoryWithZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8451BCF67F0C4F62">1.2-12 IsCategoryWithEqualizers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79D864BB79744E4E">1.2-13 IsCategoryWithCoequalizers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X82C2B6D37DA78465">1.2-14 IsEnrichedOverCommutativeRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DBCD7237CF9100A">1.2-15 IsSkeletalCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79295B4680B0F22B">1.2-16 IsAbCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X861DBAC181A15E1F">1.2-17 IsLinearCategoryOverCommutativeRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86D45F0D7804F023">1.2-18 IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7EB29BA681AE1F28">1.2-19 IsCategoryWithKernels</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8226AEFF7F6F7FC3">1.2-20 IsCategoryWithCokernels</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87C4C37D7F75E681">1.2-21 IsAdditiveCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A64E2B978685471">1.2-22 IsPreAbelianCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79DB288F84457967">1.2-23 IsAbelianCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X83EDCC4886FEE4B1">1.2-24 IsAbelianCategoryWithEnoughProjectives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X860EB7977F78EE3F">1.2-25 IsAbelianCategoryWithEnoughInjectives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87AA6D6282FD1C02">1.2-26 IsLocallyOfFiniteProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7EDD2B4386ABE96E">1.2-27 IsLocallyOfFiniteInjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87434CEF86BC38D4">1.2-28 ListOfDefiningOperations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7ABF649E831D4D8E">1.3 <span class="Heading">Constructor</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86EDB84D7E3F0194">1.3-1 CreateCapCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X854CC42E828C120D">1.3-2 CreateCapCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A1D562178FC3DDF">1.3-3 CreateCapCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8090078F7C5EF5E9">1.3-4 CreateCapCategoryWithDataTypes</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X79CE135F7E6EBFEE">1.4 <span class="Heading">Internal Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79251D5C85B0F3D3">1.4-1 Name</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C7B1EAF7D786508">1.4-2 CategoryFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BD6E2B887F86E9C">1.4-3 ObjectFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F62E0C47C778CBB">1.4-4 MorphismFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X850D0E978090D0C0">1.4-5 TwoCellFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X834BBE4085A42FAC">1.4-6 ObjectDatumType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8090D81F78B1F25A">1.4-7 MorphismDatumType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X850E7C0C807F3402">1.4-8 TwoCellDatumType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X78D0C0057817DBC3">1.4-9 CommutativeRingOfLinearCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X80AF18E4797A5BB4">1.4-10 RangeCategoryOfHomomorphismStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7CF2F93F8258DAD5">1.4-11 AdditiveGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7E094E2680348C88">1.4-12 IndecomposableProjectiveObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X85E6A5AF7DDA4F2C">1.4-13 IndecomposableInjectiveObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8305A19E86EC7B67">1.5 <span class="Heading">Logic switcher</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X82C32E8C790EAB42">1.5-1 CapCategorySwitchLogicPropagationForObjectsOn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7FEF6F218491CD45">1.5-2 CapCategorySwitchLogicPropagationForObjectsOff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X80C98B547A76705D">1.5-3 CapCategorySwitchLogicPropagationForMorphismsOn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86469EBF8261E1D4">1.5-4 CapCategorySwitchLogicPropagationForMorphismsOff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C06E84F78D82C0D">1.5-5 CapCategorySwitchLogicPropagationOn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84F114757E964E1E">1.5-6 CapCategorySwitchLogicPropagationOff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84824AC3793D872B">1.5-7 CapCategorySwitchLogicOn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84A995A687D031B7">1.5-8 CapCategorySwitchLogicOff</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A15CCB67FBCF3E3">1.6 <span class="Heading">Tool functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X851D0435816DDB40">1.6-1 CanCompute</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X820DD63180AE4BAB">1.6-2 OperationWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7AA6C5857FB663EB">1.6-3 MissingOperationsForConstructivenessOfCategory</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DB3D2BD8792ABBC">1.7 <span class="Heading">Well-Definedness of Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X839EA4D7805D4F97">1.7-1 IsWellDefined</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7D2E74CE87255099">1.8 <span class="Heading">Unpacking data structures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8790DC5F7A2BD22E">1.8-1 Down</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X858B64487B2DA04E">1.8-2 DownOnlyMorphismData</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BABD65A821C138F">1.8-3 DownToBottom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X79DB9EE785F723D1">1.9 <span class="Heading">Caching</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DB820A686B4059F">1.9-1 SetCachingOfCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87AA14F27892F14F">1.9-2 SetCachingOfCategoryWeak</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X844903657E52E6D1">1.9-3 SetDefaultCaching</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7EC631BC7CFCF5DB">1.10 <span class="Heading">Sanity checks</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BB84C7B83FE3D80">1.10-1 DisableInputSanityChecks</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7984C0E986CF0986">1.11 <span class="Heading">Timing statistics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86C6F8828628B070">1.11-1 EnableTimingStatistics</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X81A1CDD77D375231">1.12 <span class="Heading">Enable automatic calls of <code class="code">Add</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C01E3727D08D219">1.12-1 EnableAddForCategoricalOperations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7876FF1F831F58B0">1.13 <span class="Heading">Performance tweaks</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B6A66AA7D212DDC">1.14 <span class="Heading">LaTeX</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X831B157D7CC3DDA3">1.14-1 LaTeXOutput</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7B34A09D7AF0F683">1.14-2 LaTeXOutput</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap2.html#X78497E777FB3E402">2 <span class="Heading">Objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7ABD8E5A8408A301">2.1 <span class="Heading">Attributes for the Type of Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C3B8C3D865002C8">2.1-1 CapCategory</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X81817A2C84F03D1D">2.2 <span class="Heading">Adding Objects to a Category</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X824AE09A81D2B184">2.2-1 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X855C644A812F32B7">2.2-2 AddObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87BD8BB88206D940">2.2-3 CreateCapCategoryObjectWithAttributes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F5D75377B0D54A6">2.2-4 AsCapCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FAE21CF84BD7986">2.2-5 AsPrimitiveValue</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X87FC6F8A78E599E0">2.3 <span class="Heading">Equalities for Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8337569D81C0BBEC">2.3-1 IsEqualForObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C34D06C7ABE563F">2.3-2 IsIsomorphicForObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D3407D68401C878">2.3-3 SomeIsomorphismBetweenObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86CCD70884952468">2.4 <span class="Heading">Categorical Properties of Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X854468CF79122D57">2.4-1 IsBijectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X849A94147D3446BE">2.4-2 IsProjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8788D5A27EC1F994">2.4-3 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D3542EE7A8448A7">2.4-4 IsTerminal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83F6095A80DCCDC3">2.4-5 IsInitial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X856FA5E084753312">2.4-6 IsZeroForObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X824765AD78E027CC">2.4-7 IsZero</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8332653E7EA2EB72">2.5 <span class="Heading">Random Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85A6C3687C46085F">2.5-1 RandomObjectByInteger</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7EEC27307C4B06E8">2.5-2 RandomObjectByList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E0974E58188E6FB">2.5-3 RandomObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7AF3D62C7F8F40AE">2.5-4 RandomObject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7DD668627932185C">2.6 <span class="Heading">Tool functions for caches</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BAE9BBC874635CB">2.6-1 IsEqualForCacheForObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X824C0CC579E8DD7F">2.7 <span class="Heading">Object constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X825D050A8674D377">2.7-1 ObjectConstructor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C6745F27FBA23F7">2.7-2 /</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8521A36F7BBFF6BD">2.7-3 ObjectDatum</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7820B09C781A90A8">2.8 <span class="Heading">Well-Definedness of Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F9A38A187D9ABFA">2.8-1 IsWellDefinedForObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X866BE10883381E75">2.9 <span class="Heading">SetOfObjects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8047E157804C680E">2.9-1 SetOfObjectsOfCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B77708881BFE682">2.9-2 SetOfObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7EFAF5CD83C3298F">2.10 <span class="Heading">Projectives</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FED5E898784C314">2.10-1 SomeProjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D5951AA7EBD7750">2.10-2 EpimorphismFromSomeProjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81F04D8186B94570">2.10-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BFFD7FE82DCB5F6">2.10-4 ProjectiveLift</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X87100AFB7A202D2F">2.11 <span class="Heading">Injectives</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C06E1807D8775AD">2.11-1 SomeInjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8502893C82A38864">2.11-2 MonomorphismIntoSomeInjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A7C030382397A2B">2.11-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CCBB0A384F37EC6">2.11-4 InjectiveColift</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7DEC418578FAA1C0">2.12 <span class="Heading">Simplified Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BA160B382F8D742">2.12-1 Simplify</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85B54852792DEDD7">2.12-2 SimplifyObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78C0D2CB867C5C90">2.12-3 SimplifyObject_IsoFromInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X835832A685F651C8">2.12-4 SimplifyObject_IsoToInputObject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X814E7CE2792896ED">2.13 <span class="Heading">Dimensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X824604E9876DDDC5">2.13-1 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X823062CF792E3359">2.13-2 InjectiveDimension</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X7BEB6C617FED52DA">3 <span class="Heading">Morphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8642E3327E96BBDD">3.1 <span class="Heading">Attributes for the Type of Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E923036810C606E">3.1-1 CapCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E1373E385E7315A">3.1-2 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X817496B987A1ADFC">3.1-3 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83A0A39587BD6383">3.1-4 Target</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X867B9398854FEFFD">3.2 <span class="Heading">Adding Morphisms to a Category</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BDADB628232C280">3.2-1 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82453E55857562AB">3.2-2 AddMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78005D357A051FC6">3.2-3 CreateCapCategoryMorphismWithAttributes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7AE302BE7C2E829F">3.2-4 AsCapCategoryMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7893F08182290038">3.2-5 AsPrimitiveValue</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8279E00178F05A66">3.3 <span class="Heading">Morphism constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F1FB34F789ECB10">3.3-1 MorphismConstructor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BBFB178866108FA">3.3-2 MorphismDatum</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7A00D0C181471D06">3.4 <span class="Heading">Categorical Properties of Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A6F64027EEAF42B">3.4-1 IsMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E85E0A67BA11091">3.4-2 IsEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F29CF0282EFEAA3">3.4-3 IsIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X871A71747E03207D">3.4-4 IsSplitMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B4DB8C887F37D02">3.4-5 IsSplitEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FA10AFB7F441434">3.4-6 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BA4290882AFFCA3">3.4-7 IsIdempotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B655F2678F605D3">3.5 <span class="Heading">Random Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79EBA1717D89B0A4">3.5-1 RandomMorphismWithFixedSourceByInteger</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84FDBEED7844CDB6">3.5-2 RandomMorphismWithFixedSourceByList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X844B1DDC8488B7CA">3.5-3 RandomMorphismWithFixedRangeByInteger</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8568D71882C30501">3.5-4 RandomMorphismWithFixedRangeByList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80F66E9D809EA6A7">3.5-5 RandomMorphismWithFixedSourceAndRangeByInteger</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BF231C57B40B1A1">3.5-6 RandomMorphismWithFixedSourceAndRangeByList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81DB09BA7CBD3DB4">3.5-7 RandomMorphismByInteger</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E79F7A9802F622D">3.5-8 RandomMorphismByList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83D121E787FDC6E8">3.5-9 RandomMorphismWithFixedSource</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7EA50D6587A93936">3.6 <span class="Heading">Non-Categorical Properties of Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8211959783AFF307">3.6-1 IsEqualToIdentityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79D407A3807309FC">3.6-2 IsEqualToZeroMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81057B3E812B06CC">3.6-3 IsEndomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84E5DE257F241830">3.6-4 IsAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X850CFDD48773998E">3.7 <span class="Heading">Equality and Congruence for Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FABCFBF865F4179">3.7-1 IsCongruentForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8004F1DA7D9BFA8B">3.7-2 IsEqualForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8589E5A4814C3DA5">3.7-3 IsEqualForMorphismsOnMor</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8312421679CAAE35">3.8 <span class="Heading">Basic Operations for Morphisms in Ab-Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X844684AD86DCC2DC">3.8-1 IsZeroForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83B77FBB85927F14">3.8-2 AdditionForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B6461F5873F3F9D">3.8-3 SubtractionForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83177713785DCFFC">3.8-4 AdditiveInverseForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8187CAF17EB5D6D0">3.8-5 MultiplyWithElementOfCommutativeRingForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X877D74C67D041F74">3.8-6 *</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X873412DB7BD0FBD3">3.8-7 ZeroMorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C81E153817F489F">3.9 <span class="Heading">Subobject and Factorobject Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E99DAA884674DE2">3.9-1 IsEqualAsSubobjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79389FBF86CF87A4">3.9-2 IsEqualAsFactorobjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C21531C7EE65C72">3.9-3 IsDominating</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C77E9D479C8D502">3.9-4 IsCodominating</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82ADC0BF82DA2122">3.10 <span class="Heading">Identity Morphism and Composition of Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C9F3D8086238443">3.10-1 IdentityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8244F7A0868BE27D">3.10-2 PreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83CB57FB81931CFE">3.10-3 PreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87A5FFE97BB5190A">3.10-4 PreComposeList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84293B8381256A08">3.10-5 PostCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81222A1084A4F637">3.10-6 PostCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BD83BF3835870E9">3.10-7 PostComposeList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D50F754822C30D7">3.10-8 SumOfMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C101B5C7AC9089C">3.10-9 LinearCombinationOfMorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7D5182B48598FBC5">3.11 <span class="Heading">Well-Definedness of Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7968A871832877B8">3.11-1 IsWellDefinedForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X820A29637F48E932">3.11-2 IsWellDefinedForMorphismsWithGivenSourceAndRange</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83339BB37B01DB19">3.12 <span class="Heading">Lift/Colift</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83CDF65582CD9921">3.12-1 LiftAlongMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79AA9F638487EE32">3.12-2 ColiftAlongEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D9BA8D67892BD48">3.12-3 IsLiftableAlongMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DCDD2B2859EFD2D">3.12-4 IsColiftableAlongEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87C471D583C1CD9E">3.12-5 Lift</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8597DE1C83CFE89D">3.12-6 LiftOrFail</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F895375875BE5B9">3.12-7 IsLiftable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BA38B0F797AFE20">3.12-8 Colift</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79FFE43E7962CFBA">3.12-9 ColiftOrFail</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8685B04087B1E52A">3.12-10 IsColiftable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X864F84C47DC1A620">3.13 <span class="Heading">Inverses</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8371BF227B4C32EF">3.13-1 InverseForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83E4A87F7B2290F1">3.13-2 PreInverseForMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X800447917F75989A">3.13-3 PostInverseForMorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DD668627932185C">3.14 <span class="Heading">Tool functions for caches</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83F87AE07ED534ED">3.14-1 IsEqualForCacheForMorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X84563B1A835E9766">3.15 <span class="Heading">IsHomSetInhabited</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81DB15B87D1A946D">3.15-1 IsHomSetInhabited</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X858111977D44BC9C">3.16 <span class="Heading">SetOfMorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X824691307DEF08FC">3.16-1 SetOfMorphismsOfFiniteCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8426F188818F02B7">3.16-2 SetOfMorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7F065B9A78922547">3.17 <span class="Heading">Homomorphism structures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81BC29A679416C45">3.17-1 HomomorphismStructureOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8257E3787865C5E0">3.17-2 HomomorphismStructureOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FBC94B983CBA64F">3.17-3 HomomorphismStructureOnMorphismsWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7999CCFF82DA16D7">3.17-4 DistinguishedObjectOfHomomorphismStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81F1237C7DBF7603">3.17-5 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A5A05C682EB033D">3.17-6 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8760D39C865239E8">3.17-7 InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BE5E2687FC110AC">3.17-8 SolveLinearSystemInAbCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CAAF16981546CE0">3.17-9 SolveLinearSystemInAbCategoryOrFail</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A722FA5817D5F76">3.17-10 MereExistenceOfSolutionOfLinearSystemInAbCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8666962D84B81665">3.17-11 MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E6E1269854E4B94">3.17-12 MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85A7BE3185E58026">3.17-13 BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X798F78CE7EAA7F95">3.17-14 BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X860621FE7C1E6634">3.17-15 BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D473D9D83C53185">3.17-16 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7848040086B41266">3.17-17 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X834C3BB480D6561D">3.17-18 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BFD58C57EC23BD5">3.17-19 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CDC08947CF2022E">3.17-20 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X820B9AC886D78944">3.17-21 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8705DA127E089709">3.17-22 HomStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X873E91D180B3057C">3.17-23 ExtendRangeOfHomomorphismStructureByFullEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C551C8F7BA2F3F1">3.17-24 ExtendRangeOfHomomorphismStructureByIdentityAsFullEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8510A7E08672FD73">3.17-25 MorphismsOfExternalHom</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CC9C1A487AE6086">3.17-26 BasisOfExternalHom</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8636593C845B23D2">3.17-27 CoefficientsOfMorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B39E2927B533D10">3.18 <span class="Heading">Simplified Morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FCE611883F6A56B">3.18-1 SimplifyMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DAD0E437D28F300">3.18-2 SimplifySource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X843C68B97ED31DD4">3.18-3 SimplifySource_IsoToInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CE5C1BB7BA7AA16">3.18-4 SimplifySource_IsoFromInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F80C5357F84A39A">3.18-5 SimplifyRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82FCEC277F3A1A70">3.18-6 SimplifyRange_IsoToInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X865CD95E7D068A77">3.18-7 SimplifyRange_IsoFromInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X821E16DF8071B8A4">3.18-8 SimplifySourceAndRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84C39DE584D38D4E">3.18-9 SimplifySourceAndRange_IsoToInputRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X859DF3C2819FFE4D">3.18-10 SimplifySourceAndRange_IsoFromInputRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DD0915A7E23283E">3.18-11 SimplifySourceAndRange_IsoToInputSource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X783C20A47889C4A8">3.18-12 SimplifySourceAndRange_IsoFromInputSource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X830C7337864DB3D6">3.18-13 SimplifyEndo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78189F508549ABAA">3.18-14 SimplifyEndo_IsoToInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C07985D844FD715">3.18-15 SimplifyEndo_IsoFromInputObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D80D2B2792A7A50">3.18-16 Simplify</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8379B7817AA67F2D">3.19 <span class="Heading">Reduction by split epi summands</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F997AF37E59411F">3.19-1 SomeReductionBySplitEpiSummand</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C92D8AF82E65289">3.19-2 SomeReductionBySplitEpiSummand_MorphismToInputRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A0F381279955981">3.19-3 SomeReductionBySplitEpiSummand_MorphismFromInputRange</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X7B2650D983390045">4 <span class="Heading">Category 2-Cells</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X793836877FCD5EC6">4.1 <span class="Heading">Attributes for the Type of 2-Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8061EB36809022A4">4.1-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C16D44B8639016B">4.1-2 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X780D00C482CA707D">4.1-3 Target</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7CB51AC67E08E3A1">4.2 <span class="Heading">Adding 2-Cells to a Category</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82FFCA6D82FC0ED9">4.2-1 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E3D6C538354F1D9">4.2-2 AddTwoCell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B449CCB7C8050A3">4.2-3 CreateCapCategoryTwoCellWithAttributes</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B780EDB7E466121">4.3 <span class="Heading">Identity 2-Cell and Composition of 2-Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78ED1F327D902924">4.3-1 IdentityTwoCell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7DD9EB1F80921939">4.3-2 HorizontalPreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87C942B87C6A161D">4.3-3 HorizontalPostCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D89385B7834B086">4.3-4 VerticalPreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79D9FBEA851C4F00">4.3-5 VerticalPostCompose</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X84A60F017943A943">4.4 <span class="Heading">Well-Definedness for 2-Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8240AF367C87663F">4.4-1 IsWellDefinedForTwoCells</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X87C0818C7EFB273D">5 <span class="Heading">Category of Categories</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7A8C280B85B50DE0">5.1 <span class="Heading">The Category Cat</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X810DBB407D117E6B">5.1-1 CapCat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CC6903E78F24167">5.2 <span class="Heading">Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X814369D484E429EF">5.2-1 IsCapCategoryAsCatObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CBE360B8629DC79">5.2-2 IsCapFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D2DD1297D6E57D0">5.2-3 IsCapNaturalTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X86EC0F0A78ECBC10">5.3 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7F7E47537FD1EDC3">5.3-1 AsCatObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E3142BA84C1EC72">5.3-2 AsCapCategory</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X78D1062D78BE08C1">5.4 <span class="Heading">Functors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7814A6FC86238C93">5.4-1 CapFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E4F877C7C91B7CA">5.4-2 CapFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78773EEC7F33AC3A">5.4-3 SourceOfFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8710DBB68544497A">5.4-4 RangeOfFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A75B6A77C9F6836">5.4-5 AddObjectFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78CD5D1E811A85CF">5.4-6 FunctorObjectOperation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X85EFB9BE8145B737">5.4-7 AddMorphismFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D3BAF84793D9576">5.4-8 FunctorMorphismOperation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X850397447A1A766E">5.4-9 ApplyFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8562CA1F795C5154">5.4-10 InputSignature</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X781664C7876ED8A0">5.4-11 InstallFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X833F9B6D85713A4F">5.4-12 IdentityFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83FFF5538147E243">5.4-13 FunctorCanonicalizeZeroObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EF63850830B8064">5.4-14 NaturalIsomorphismFromIdentityToCanonicalizeZeroObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7AA6B22B7BF73AAE">5.4-15 FunctorCanonicalizeZeroMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8223CAD483193F52">5.4-16 NaturalIsomorphismFromIdentityToCanonicalizeZeroMorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X836749D8814FEEE6">5.5 <span class="Heading">Natural transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7F32E8C985178781">5.5-1 Name</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C9810B8858249A0">5.5-2 NaturalTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DFC4C9183446E95">5.5-3 AddNaturalTransformationFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X867F8B6A78462396">5.5-4 ApplyNaturalTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X79E6BE7E83687767">5.5-5 InstallNaturalTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EAB133B81CDB5E5">5.5-6 HorizontalPreComposeNaturalTransformationWithFunctor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B8AC0BB8370031D">5.5-7 HorizontalPreComposeFunctorWithNaturalTransformation</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X7994CC1487D7617C">6 <span class="Heading">Universal Objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7DCD99628504B810">6.1 <span class="Heading">Kernel</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82EAD3357C9FE4C8">6.1-1 KernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8430666980D732FB">6.1-2 KernelEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CBE1D0C84F5A47E">6.1-3 KernelEmbeddingWithGivenKernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78140AF884DE0736">6.1-4 MorphismFromKernelObjectToSink</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7BF5A3A57E23C795">6.1-5 MorphismFromKernelObjectToSinkWithGivenKernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87920AC67A5802BC">6.1-6 KernelLift</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83D439A07AD29D05">6.1-7 KernelLiftWithGivenKernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78D5B94A7EF4D4F0">6.1-8 KernelObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X787AE8F07F7A062C">6.1-9 KernelObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7F61FE0B82AF4CDB">6.1-10 KernelObjectFunctorialWithGivenKernelObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A17922D869CCEB5">6.1-11 KernelObjectFunctorialWithGivenKernelObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X875F177A82BF9B8B">6.2 <span class="Heading">Cokernel</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82803DBC80F40EFC">6.2-1 CokernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78948D7A7B52AB31">6.2-2 CokernelProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X845DC9377A05BE7B">6.2-3 CokernelProjectionWithGivenCokernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X872E26C57F195D50">6.2-4 MorphismFromSourceToCokernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X876648527E75AA04">6.2-5 MorphismFromSourceToCokernelObjectWithGivenCokernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8735262382CE2560">6.2-6 CokernelColift</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86DB5CD27C0EB7D2">6.2-7 CokernelColiftWithGivenCokernelObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78E421227FB90A70">6.2-8 CokernelObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X837889ED7BD6CBED">6.2-9 CokernelObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7BA172F07A431626">6.2-10 CokernelObjectFunctorialWithGivenCokernelObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7F59A455873709D3">6.2-11 CokernelObjectFunctorialWithGivenCokernelObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X797A74DE8267C974">6.3 <span class="Heading">Zero Object</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X790C4FA87CADB93E">6.3-1 ZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87D87A177FE0542F">6.3-2 ZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A0BF1118777C8A3">6.3-3 UniversalMorphismFromZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X80327B2387F38FE8">6.3-4 UniversalMorphismFromZeroObjectWithGivenZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86003308844F8341">6.3-5 UniversalMorphismIntoZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X866104CA84CBC40A">6.3-6 UniversalMorphismIntoZeroObjectWithGivenZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7F701A11812C74C5">6.3-7 MorphismFromZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X837BD808791003FF">6.3-8 MorphismIntoZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8711ABE9811C7CCF">6.3-9 IsomorphismFromZeroObjectToInitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86FF82B284B8E2EB">6.3-10 IsomorphismFromInitialObjectToZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78C501A179BB6CBB">6.3-11 IsomorphismFromZeroObjectToTerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83E84FC187EE2445">6.3-12 IsomorphismFromTerminalObjectToZeroObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X79CEAF827DDED44B">6.3-13 ZeroObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82D7BF1F85268B0A">6.3-14 ZeroObjectFunctorialWithGivenZeroObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X827CD17C7EBFD58F">6.4 <span class="Heading">Terminal Object</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7DC837217946D22D">6.4-1 TerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7D86D2EA7845AEEB">6.4-2 TerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7BEA5AF67D63F4A5">6.4-3 UniversalMorphismIntoTerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8707AD1784DCBBFF">6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C616CD287760D2F">6.4-5 TerminalObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8511229882130F93">6.4-6 TerminalObjectFunctorialWithGivenTerminalObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X78B0ED8B80BF5254">6.5 <span class="Heading">Initial Object</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A70384E7F182B00">6.5-1 InitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7E17CDF481C348B9">6.5-2 InitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X873FC2B087004DC3">6.5-3 UniversalMorphismFromInitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7F7177F585576F6B">6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87B1C71179F798C8">6.5-5 InitialObjectFunctorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CE9BBC27F70F3BD">6.5-6 InitialObjectFunctorialWithGivenInitialObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X81FDB99378D1307A">6.6 <span class="Heading">Direct Sum</span></a>
</span>
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