  
  
                                     [1X CAP [101X
  
  
                     [1X Categories, Algorithms, Programming [101X
  
  
                                   2020.10-01
  
  
                                19 October 2020
  
  
                               Sebastian Gutsche
  
                                Sebastian Posur
  
                            Øystein Skartsæterhagen
  
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:gutsche@mathematik.uni-siegen.de[107X
      Homepage: [7Xhttps://sebasguts.github.io/[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Sebastian Posur
      Email:    [7Xmailto:sebastian.posur@uni-siegen.de[107X
      Homepage: [7Xhttps://sebastianpos.github.io[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Øystein Skartsæterhagen
      Email:    [7Xmailto:oysteini@math.ntnu.no[107X
      Homepage: [7Xhttp://www.math.ntnu.no/~oysteini/[107X
      Address:  [33X[0;14YNTNU[133X
                [33X[0;14YInstitutt for matematiske fag[133X
                [33X[0;14Y7491 Trondheim[133X
                [33X[0;14YNorway[133X
  
  
  
  -------------------------------------------------------
  
  
  [1XContents (CAP)[101X
  
  1 [33X[0;0YCAP Categories[133X
    1.1 [33X[0;0YCategories[133X
      1.1-1 IsCapCategory
      1.1-2 IsCapCategoryCell
      1.1-3 IsCapCategoryObject
      1.1-4 IsCapCategoryMorphism
      1.1-5 IsCapCategoryTwoCell
      1.1-6 AddCategoricalProperty
    1.2 [33X[0;0YConstructor[133X
      1.2-1 CreateCapCategory
      1.2-2 CreateCapCategory
    1.3 [33X[0;0YInternal Attributes[133X
      1.3-1 CategoryFilter
      1.3-2 CellFilter
      1.3-3 ObjectFilter
      1.3-4 MorphismFilter
      1.3-5 TwoCellFilter
      1.3-6 CommutativeRingOfLinearCategory
    1.4 [33X[0;0YLogic switcher[133X
      1.4-1 CapCategorySwitchLogicPropagationForObjectsOn
      1.4-2 CapCategorySwitchLogicPropagationForObjectsOff
      1.4-3 CapCategorySwitchLogicPropagationForMorphismsOn
      1.4-4 CapCategorySwitchLogicPropagationForMorphismsOff
      1.4-5 CapCategorySwitchLogicPropagationOn
      1.4-6 CapCategorySwitchLogicPropagationOff
      1.4-7 CapCategorySwitchLogicOn
      1.4-8 CapCategorySwitchLogicOff
    1.5 [33X[0;0YTool functions[133X
      1.5-1 CanCompute
      1.5-2 CheckConstructivenessOfCategory
    1.6 [33X[0;0YWell-Definedness of Cells[133X
      1.6-1 IsWellDefined
    1.7 [33X[0;0YUnpacking data structures[133X
      1.7-1 Down
      1.7-2 DownOnlyMorphismData
      1.7-3 DownToBottom
    1.8 [33X[0;0YCaching[133X
      1.8-1 SetCachingOfCategory
      1.8-2 SetCachingOfCategoryWeak
      1.8-3 SetDefaultCaching
    1.9 [33X[0;0YSanity checks[133X
      1.9-1 DisableInputSanityChecks
    1.10 [33X[0;0YEnable automatic calls of [10XAdd[110X[133X
      1.10-1 EnableAddForCategoricalOperations
    1.11 [33X[0;0YPerformance tweaks[133X
    1.12 [33X[0;0YLaTeX[133X
      1.12-1 LaTeXOutput
      1.12-2 LaTeXOutput
  2 [33X[0;0YObjects[133X
    2.1 [33X[0;0YAttributes for the Type of Objects[133X
      2.1-1 CapCategory
    2.2 [33X[0;0YEquality for Objects[133X
      2.2-1 IsEqualForObjects
      2.2-2 AddIsEqualForObjects
    2.3 [33X[0;0YCategorical Properties of Objects[133X
      2.3-1 IsProjective
      2.3-2 AddIsProjective
      2.3-3 IsInjective
      2.3-4 AddIsInjective
      2.3-5 IsTerminal
      2.3-6 AddIsTerminal
      2.3-7 IsInitial
      2.3-8 AddIsInitial
      2.3-9 IsZeroForObjects
      2.3-10 IsZero
      2.3-11 AddIsZeroForObjects
    2.4 [33X[0;0YRandom Objects[133X
      2.4-1 RandomObjectByInteger
      2.4-2 AddRandomObjectByInteger
      2.4-3 RandomObjectByList
      2.4-4 AddRandomObjectByList
      2.4-5 RandomObject
    2.5 [33X[0;0YTool functions for caches[133X
      2.5-1 IsEqualForCacheForObjects
      2.5-2 AddIsEqualForCacheForObjects
    2.6 [33X[0;0YAdding Objects to a Category[133X
      2.6-1 Add
      2.6-2 AddObject
      2.6-3 AddObjectRepresentation
      2.6-4 ObjectifyObjectForCAPWithAttributes
    2.7 [33X[0;0YWell-Definedness of Objects[133X
      2.7-1 IsWellDefinedForObjects
      2.7-2 AddIsWellDefinedForObjects
    2.8 [33X[0;0YProjectives[133X
      2.8-1 SomeProjectiveObject
      2.8-2 EpimorphismFromSomeProjectiveObject
      2.8-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
      2.8-4 ProjectiveLift
      2.8-5 AddSomeProjectiveObject
      2.8-6 AddEpimorphismFromSomeProjectiveObject
      2.8-7 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
      2.8-8 AddProjectiveLift
    2.9 [33X[0;0YInjectives[133X
      2.9-1 SomeInjectiveObject
      2.9-2 MonomorphismIntoSomeInjectiveObject
      2.9-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
      2.9-4 InjectiveColift
      2.9-5 AddSomeInjectiveObject
      2.9-6 AddMonomorphismIntoSomeInjectiveObject
      2.9-7 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
      2.9-8 AddInjectiveColift
    2.10 [33X[0;0YSimplified Objects[133X
      2.10-1 Simplify
      2.10-2 SimplifyObject
      2.10-3 AddSimplifyObject
      2.10-4 SimplifyObject_IsoFromInputObject
      2.10-5 AddSimplifyObject_IsoFromInputObject
      2.10-6 SimplifyObject_IsoToInputObject
      2.10-7 AddSimplifyObject_IsoToInputObject
  3 [33X[0;0YMorphisms[133X
    3.1 [33X[0;0YAttributes for the Type of Morphisms[133X
      3.1-1 CapCategory
      3.1-2 Source
      3.1-3 Range
    3.2 [33X[0;0YCategorical Properties of Morphisms[133X
      3.2-1 IsMonomorphism
      3.2-2 AddIsMonomorphism
      3.2-3 IsEpimorphism
      3.2-4 AddIsEpimorphism
      3.2-5 IsIsomorphism
      3.2-6 AddIsIsomorphism
      3.2-7 IsSplitMonomorphism
      3.2-8 AddIsSplitMonomorphism
      3.2-9 IsSplitEpimorphism
      3.2-10 AddIsSplitEpimorphism
      3.2-11 IsOne
      3.2-12 AddIsOne
      3.2-13 IsIdempotent
      3.2-14 AddIsIdempotent
    3.3 [33X[0;0YRandom Morphisms[133X
      3.3-1 RandomMorphismWithFixedSourceByInteger
      3.3-2 AddRandomMorphismWithFixedSourceByInteger
      3.3-3 RandomMorphismWithFixedSourceByList
      3.3-4 AddRandomMorphismWithFixedSourceByList
      3.3-5 RandomMorphismWithFixedRangeByInteger
      3.3-6 AddRandomMorphismWithFixedRangeByInteger
      3.3-7 RandomMorphismWithFixedRangeByList
      3.3-8 AddRandomMorphismWithFixedRangeByList
      3.3-9 RandomMorphismWithFixedSourceAndRangeByInteger
      3.3-10 AddRandomMorphismWithFixedSourceAndRangeByInteger
      3.3-11 RandomMorphismWithFixedSourceAndRangeByList
      3.3-12 AddRandomMorphismWithFixedSourceAndRangeByList
      3.3-13 RandomMorphismByInteger
      3.3-14 AddRandomMorphismByInteger
      3.3-15 RandomMorphismByList
      3.3-16 AddRandomMorphismByList
      3.3-17 RandomMorphismWithFixedSource
    3.4 [33X[0;0YNon-Categorical Properties of Morphisms[133X
      3.4-1 IsIdenticalToIdentityMorphism
      3.4-2 AddIsIdenticalToIdentityMorphism
      3.4-3 IsIdenticalToZeroMorphism
      3.4-4 AddIsIdenticalToZeroMorphism
      3.4-5 IsEndomorphism
      3.4-6 AddIsEndomorphism
      3.4-7 IsAutomorphism
      3.4-8 AddIsAutomorphism
    3.5 [33X[0;0YAdding Morphisms to a Category[133X
      3.5-1 Add
      3.5-2 AddMorphism
      3.5-3 AddMorphismRepresentation
      3.5-4 ObjectifyMorphismWithSourceAndRangeForCAPWithAttributes
    3.6 [33X[0;0YEquality and Congruence for Morphisms[133X
      3.6-1 IsCongruentForMorphisms
      3.6-2 AddIsCongruentForMorphisms
      3.6-3 IsEqualForMorphisms
      3.6-4 AddIsEqualForMorphisms
      3.6-5 IsEqualForMorphismsOnMor
      3.6-6 AddIsEqualForMorphismsOnMor
    3.7 [33X[0;0YBasic Operations for Morphisms in Ab-Categories[133X
      3.7-1 IsZeroForMorphisms
      3.7-2 AddIsZeroForMorphisms
      3.7-3 AdditionForMorphisms
      3.7-4 AddAdditionForMorphisms
      3.7-5 SubtractionForMorphisms
      3.7-6 AddSubtractionForMorphisms
      3.7-7 AdditiveInverseForMorphisms
      3.7-8 AddAdditiveInverseForMorphisms
      3.7-9 MultiplyWithElementOfCommutativeRingForMorphisms
      3.7-10 \*
      3.7-11 AddMultiplyWithElementOfCommutativeRingForMorphisms
      3.7-12 ZeroMorphism
      3.7-13 AddZeroMorphism
    3.8 [33X[0;0YSubobject and Factorobject Operations[133X
      3.8-1 IsEqualAsSubobjects
      3.8-2 AddIsEqualAsSubobjects
      3.8-3 IsEqualAsFactorobjects
      3.8-4 AddIsEqualAsFactorobjects
      3.8-5 IsDominating
      3.8-6 AddIsDominating
      3.8-7 IsCodominating
      3.8-8 AddIsCodominating
    3.9 [33X[0;0YIdentity Morphism and Composition of Morphisms[133X
      3.9-1 IdentityMorphism
      3.9-2 AddIdentityMorphism
      3.9-3 PreCompose
      3.9-4 PreCompose
      3.9-5 AddPreCompose
      3.9-6 PostCompose
      3.9-7 PostCompose
      3.9-8 AddPostCompose
    3.10 [33X[0;0YWell-Definedness of Morphisms[133X
      3.10-1 IsWellDefinedForMorphisms
      3.10-2 AddIsWellDefinedForMorphisms
    3.11 [33X[0;0YLift/Colift[133X
      3.11-1 LiftAlongMonomorphism
      3.11-2 AddLiftAlongMonomorphism
      3.11-3 ColiftAlongEpimorphism
      3.11-4 AddColiftAlongEpimorphism
      3.11-5 IsLiftableAlongMonomorphism
      3.11-6 AddIsLiftableAlongMonomorphism
      3.11-7 IsColiftableAlongEpimorphism
      3.11-8 AddIsColiftableAlongEpimorphism
      3.11-9 Lift
      3.11-10 AddLift
      3.11-11 Colift
      3.11-12 AddColift
      3.11-13 IsLiftable
      3.11-14 AddIsLiftable
      3.11-15 IsColiftable
      3.11-16 AddIsColiftable
    3.12 [33X[0;0YInverses[133X
      3.12-1 Inverse
      3.12-2 AddInverse
    3.13 [33X[0;0YTool functions for caches[133X
      3.13-1 IsEqualForCacheForMorphisms
      3.13-2 AddIsEqualForCacheForMorphisms
    3.14 [33X[0;0YIsHomSetInhabited[133X
      3.14-1 IsHomSetInhabited
      3.14-2 AddIsHomSetInhabited
    3.15 [33X[0;0YHomomorphism structures[133X
      3.15-1 HomomorphismStructureOnObjects
      3.15-2 AddHomomorphismStructureOnObjects
      3.15-3 HomomorphismStructureOnMorphisms
      3.15-4 HomomorphismStructureOnMorphismsWithGivenObjects
      3.15-5 AddHomomorphismStructureOnMorphismsWithGivenObjects
      3.15-6 DistinguishedObjectOfHomomorphismStructure
      3.15-7 AddDistinguishedObjectOfHomomorphismStructure
      3.15-8 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure
      3.15-9 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure
      3.15-10 InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
      3.15-11 AddInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
      3.15-12 SolveLinearSystemInAbCategory
      3.15-13 MereExistenceOfSolutionOfLinearSystemInAbCategory
      3.15-14 HomStructure
      3.15-15 HomStructure
      3.15-16 HomStructure
      3.15-17 HomStructure
      3.15-18 HomStructure
      3.15-19 HomStructure
      3.15-20 HomStructure
      3.15-21 BasisOfExternalHom
      3.15-22 AddBasisOfExternalHom
      3.15-23 CoefficientsOfMorphismWithGivenBasisOfExternalHom
      3.15-24 AddCoefficientsOfMorphismWithGivenBasisOfExternalHom
      3.15-25 CoefficientsOfMorphism
    3.16 [33X[0;0YSimplified Morphisms[133X
      3.16-1 SimplifyMorphism
      3.16-2 AddSimplifyMorphism
      3.16-3 SimplifySource
      3.16-4 AddSimplifySource
      3.16-5 SimplifySource_IsoToInputObject
      3.16-6 AddSimplifySource_IsoToInputObject
      3.16-7 SimplifySource_IsoFromInputObject
      3.16-8 AddSimplifySource_IsoFromInputObject
      3.16-9 SimplifyRange
      3.16-10 AddSimplifyRange
      3.16-11 SimplifyRange_IsoToInputObject
      3.16-12 AddSimplifyRange_IsoToInputObject
      3.16-13 SimplifyRange_IsoFromInputObject
      3.16-14 AddSimplifyRange_IsoFromInputObject
      3.16-15 SimplifySourceAndRange
      3.16-16 AddSimplifySourceAndRange
      3.16-17 SimplifySourceAndRange_IsoToInputRange
      3.16-18 AddSimplifySourceAndRange_IsoToInputRange
      3.16-19 SimplifySourceAndRange_IsoFromInputRange
      3.16-20 AddSimplifySourceAndRange_IsoFromInputRange
      3.16-21 SimplifySourceAndRange_IsoToInputSource
      3.16-22 AddSimplifySourceAndRange_IsoToInputSource
      3.16-23 SimplifySourceAndRange_IsoFromInputSource
      3.16-24 AddSimplifySourceAndRange_IsoFromInputSource
      3.16-25 SimplifyEndo
      3.16-26 AddSimplifyEndo
      3.16-27 SimplifyEndo_IsoToInputObject
      3.16-28 AddSimplifyEndo_IsoToInputObject
      3.16-29 SimplifyEndo_IsoFromInputObject
      3.16-30 AddSimplifyEndo_IsoFromInputObject
      3.16-31 Simplify
    3.17 [33X[0;0YReduction by split epi summands[133X
      3.17-1 SomeReductionBySplitEpiSummand
      3.17-2 AddSomeReductionBySplitEpiSummand
      3.17-3 SomeReductionBySplitEpiSummand_MorphismToInputRange
      3.17-4 AddSomeReductionBySplitEpiSummand_MorphismToInputRange
      3.17-5 SomeReductionBySplitEpiSummand_MorphismFromInputRange
      3.17-6 AddSomeReductionBySplitEpiSummand_MorphismFromInputRange
  4 [33X[0;0YCategory 2-Cells[133X
    4.1 [33X[0;0YAttributes for the Type of 2-Cells[133X
      4.1-1 Source
      4.1-2 Range
    4.2 [33X[0;0YIdentity 2-Cell and Composition of 2-Cells[133X
      4.2-1 IdentityTwoCell
      4.2-2 AddIdentityTwoCell
      4.2-3 HorizontalPreCompose
      4.2-4 AddHorizontalPreCompose
      4.2-5 HorizontalPostCompose
      4.2-6 AddHorizontalPostCompose
      4.2-7 VerticalPreCompose
      4.2-8 AddVerticalPreCompose
      4.2-9 VerticalPostCompose
      4.2-10 AddVerticalPostCompose
    4.3 [33X[0;0YWell-Definedness for 2-Cells[133X
      4.3-1 IsWellDefinedForTwoCells
      4.3-2 AddIsWellDefinedForTwoCells
  5 [33X[0;0YCategory of Categories[133X
    5.1 [33X[0;0YThe Category Cat[133X
      5.1-1 CapCat
    5.2 [33X[0;0YCategories[133X
      5.2-1 IsCapCategoryAsCatObject
      5.2-2 IsCapFunctor
      5.2-3 IsCapNaturalTransformation
    5.3 [33X[0;0YConstructors[133X
      5.3-1 AsCatObject
      5.3-2 AsCapCategory
    5.4 [33X[0;0YFunctors[133X
      5.4-1 CapFunctor
      5.4-2 CapFunctor
      5.4-3 SourceOfFunctor
      5.4-4 RangeOfFunctor
      5.4-5 AddObjectFunction
      5.4-6 FunctorObjectOperation
      5.4-7 AddMorphismFunction
      5.4-8 FunctorMorphismOperation
      5.4-9 ApplyFunctor
      5.4-10 InputSignature
      5.4-11 InstallFunctor
      5.4-12 IdentityFunctor
      5.4-13 FunctorCanonicalizeZeroObjects
      5.4-14 NaturalIsomorphismFromIdentityToCanonicalizeZeroObjects
      5.4-15 FunctorCanonicalizeZeroMorphisms
      5.4-16 NaturalIsomorphismFromIdentityToCanonicalizeZeroMorphisms
    5.5 [33X[0;0YNatural transformations[133X
      5.5-1 Name
      5.5-2 NaturalTransformation
      5.5-3 AddNaturalTransformationFunction
      5.5-4 ApplyNaturalTransformation
      5.5-5 InstallNaturalTransformation
      5.5-6 HorizontalPreComposeNaturalTransformationWithFunctor
      5.5-7 HorizontalPreComposeFunctorWithNaturalTransformation
  6 [33X[0;0YUniversal Objects[133X
    6.1 [33X[0;0YKernel[133X
      6.1-1 KernelObject
      6.1-2 KernelEmbedding
      6.1-3 KernelEmbeddingWithGivenKernelObject
      6.1-4 MorphismFromKernelObjectToSink
      6.1-5 MorphismFromKernelObjectToSinkWithGivenKernelObject
      6.1-6 KernelLift
      6.1-7 KernelLiftWithGivenKernelObject
      6.1-8 AddKernelObject
      6.1-9 AddKernelEmbedding
      6.1-10 AddKernelEmbeddingWithGivenKernelObject
      6.1-11 AddMorphismFromKernelObjectToSink
      6.1-12 AddMorphismFromKernelObjectToSinkWithGivenKernelObject
      6.1-13 AddKernelLift
      6.1-14 AddKernelLiftWithGivenKernelObject
      6.1-15 KernelObjectFunctorial
      6.1-16 KernelObjectFunctorial
      6.1-17 KernelObjectFunctorialWithGivenKernelObjects
      6.1-18 KernelObjectFunctorialWithGivenKernelObjects
      6.1-19 AddKernelObjectFunctorialWithGivenKernelObjects
    6.2 [33X[0;0YCokernel[133X
      6.2-1 CokernelObject
      6.2-2 CokernelProjection
      6.2-3 CokernelProjectionWithGivenCokernelObject
      6.2-4 MorphismFromSourceToCokernelObject
      6.2-5 MorphismFromSourceToCokernelObjectWithGivenCokernelObject
      6.2-6 CokernelColift
      6.2-7 CokernelColiftWithGivenCokernelObject
      6.2-8 AddCokernelObject
      6.2-9 AddCokernelProjection
      6.2-10 AddCokernelProjectionWithGivenCokernelObject
      6.2-11 AddMorphismFromSourceToCokernelObject
      6.2-12 AddMorphismFromSourceToCokernelObjectWithGivenCokernelObject
      6.2-13 AddCokernelColift
      6.2-14 AddCokernelColiftWithGivenCokernelObject
      6.2-15 CokernelObjectFunctorial
      6.2-16 CokernelObjectFunctorial
      6.2-17 CokernelObjectFunctorialWithGivenCokernelObjects
      6.2-18 CokernelObjectFunctorialWithGivenCokernelObjects
      6.2-19 AddCokernelObjectFunctorialWithGivenCokernelObjects
    6.3 [33X[0;0YZero Object[133X
      6.3-1 ZeroObject
      6.3-2 ZeroObject
      6.3-3 MorphismFromZeroObject
      6.3-4 MorphismIntoZeroObject
      6.3-5 UniversalMorphismFromZeroObject
      6.3-6 UniversalMorphismFromZeroObjectWithGivenZeroObject
      6.3-7 UniversalMorphismIntoZeroObject
      6.3-8 UniversalMorphismIntoZeroObjectWithGivenZeroObject
      6.3-9 IsomorphismFromZeroObjectToInitialObject
      6.3-10 IsomorphismFromInitialObjectToZeroObject
      6.3-11 IsomorphismFromZeroObjectToTerminalObject
      6.3-12 IsomorphismFromTerminalObjectToZeroObject
      6.3-13 AddZeroObject
      6.3-14 AddUniversalMorphismIntoZeroObject
      6.3-15 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject
      6.3-16 AddUniversalMorphismFromZeroObject
      6.3-17 AddUniversalMorphismFromZeroObjectWithGivenZeroObject
      6.3-18 AddIsomorphismFromZeroObjectToInitialObject
      6.3-19 AddIsomorphismFromInitialObjectToZeroObject
      6.3-20 AddIsomorphismFromZeroObjectToTerminalObject
      6.3-21 AddIsomorphismFromTerminalObjectToZeroObject
      6.3-22 ZeroObjectFunctorial
      6.3-23 AddZeroObjectFunctorial
    6.4 [33X[0;0YTerminal Object[133X
      6.4-1 TerminalObject
      6.4-2 TerminalObject
      6.4-3 UniversalMorphismIntoTerminalObject
      6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject
      6.4-5 AddTerminalObject
      6.4-6 AddUniversalMorphismIntoTerminalObject
      6.4-7 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject
      6.4-8 TerminalObjectFunctorial
      6.4-9 AddTerminalObjectFunctorial
    6.5 [33X[0;0YInitial Object[133X
      6.5-1 InitialObject
      6.5-2 InitialObject
      6.5-3 UniversalMorphismFromInitialObject
      6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject
      6.5-5 AddInitialObject
      6.5-6 AddUniversalMorphismFromInitialObject
      6.5-7 AddUniversalMorphismFromInitialObjectWithGivenInitialObject
      6.5-8 InitialObjectFunctorial
      6.5-9 AddInitialObjectFunctorial
    6.6 [33X[0;0YDirect Sum[133X
      6.6-1 DirectSumOp
      6.6-2 ProjectionInFactorOfDirectSum
      6.6-3 ProjectionInFactorOfDirectSumOp
      6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum
      6.6-5 InjectionOfCofactorOfDirectSum
      6.6-6 InjectionOfCofactorOfDirectSumOp
      6.6-7 InjectionOfCofactorOfDirectSumWithGivenDirectSum
      6.6-8 UniversalMorphismIntoDirectSum
      6.6-9 UniversalMorphismIntoDirectSumOp
      6.6-10 UniversalMorphismIntoDirectSumWithGivenDirectSum
      6.6-11 UniversalMorphismFromDirectSum
      6.6-12 UniversalMorphismFromDirectSumOp
      6.6-13 UniversalMorphismFromDirectSumWithGivenDirectSum
      6.6-14 IsomorphismFromDirectSumToDirectProduct
      6.6-15 IsomorphismFromDirectSumToDirectProductOp
      6.6-16 IsomorphismFromDirectProductToDirectSum
      6.6-17 IsomorphismFromDirectProductToDirectSumOp
      6.6-18 IsomorphismFromDirectSumToCoproduct
      6.6-19 IsomorphismFromDirectSumToCoproductOp
      6.6-20 IsomorphismFromCoproductToDirectSum
      6.6-21 IsomorphismFromCoproductToDirectSumOp
      6.6-22 MorphismBetweenDirectSums
      6.6-23 AddMorphismBetweenDirectSums
      6.6-24 MorphismBetweenDirectSumsOp
      6.6-25 ComponentOfMorphismIntoDirectSum
      6.6-26 ComponentOfMorphismFromDirectSum
      6.6-27 AddComponentOfMorphismIntoDirectSum
      6.6-28 AddComponentOfMorphismFromDirectSum
      6.6-29 AddProjectionInFactorOfDirectSum
      6.6-30 AddProjectionInFactorOfDirectSumWithGivenDirectSum
      6.6-31 AddInjectionOfCofactorOfDirectSum
      6.6-32 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum
      6.6-33 AddUniversalMorphismIntoDirectSum
      6.6-34 AddUniversalMorphismIntoDirectSumWithGivenDirectSum
      6.6-35 AddUniversalMorphismFromDirectSum
      6.6-36 AddUniversalMorphismFromDirectSumWithGivenDirectSum
      6.6-37 AddIsomorphismFromDirectSumToDirectProduct
      6.6-38 AddIsomorphismFromDirectProductToDirectSum
      6.6-39 AddIsomorphismFromDirectSumToCoproduct
      6.6-40 AddIsomorphismFromCoproductToDirectSum
      6.6-41 AddDirectSum
      6.6-42 DirectSumFunctorial
      6.6-43 DirectSumFunctorialWithGivenDirectSums
      6.6-44 AddDirectSumFunctorialWithGivenDirectSums
    6.7 [33X[0;0YCoproduct[133X
      6.7-1 Coproduct
      6.7-2 Coproduct
      6.7-3 Coproduct
      6.7-4 CoproductOp
      6.7-5 InjectionOfCofactorOfCoproduct
      6.7-6 InjectionOfCofactorOfCoproductOp
      6.7-7 InjectionOfCofactorOfCoproductWithGivenCoproduct
      6.7-8 UniversalMorphismFromCoproduct
      6.7-9 UniversalMorphismFromCoproductOp
      6.7-10 UniversalMorphismFromCoproductWithGivenCoproduct
      6.7-11 AddCoproduct
      6.7-12 AddInjectionOfCofactorOfCoproduct
      6.7-13 AddInjectionOfCofactorOfCoproductWithGivenCoproduct
      6.7-14 AddUniversalMorphismFromCoproduct
      6.7-15 AddUniversalMorphismFromCoproductWithGivenCoproduct
      6.7-16 CoproductFunctorial
      6.7-17 CoproductFunctorialWithGivenCoproducts
      6.7-18 AddCoproductFunctorialWithGivenCoproducts
    6.8 [33X[0;0YDirect Product[133X
      6.8-1 DirectProductOp
      6.8-2 ProjectionInFactorOfDirectProduct
      6.8-3 ProjectionInFactorOfDirectProductOp
      6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct
      6.8-5 UniversalMorphismIntoDirectProduct
      6.8-6 UniversalMorphismIntoDirectProductOp
      6.8-7 UniversalMorphismIntoDirectProductWithGivenDirectProduct
      6.8-8 AddDirectProduct
      6.8-9 AddProjectionInFactorOfDirectProduct
      6.8-10 AddProjectionInFactorOfDirectProductWithGivenDirectProduct
      6.8-11 AddUniversalMorphismIntoDirectProduct
      6.8-12 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct
      6.8-13 DirectProductFunctorial
      6.8-14 DirectProductFunctorialWithGivenDirectProducts
      6.8-15 AddDirectProductFunctorialWithGivenDirectProducts
    6.9 [33X[0;0YEqualizer[133X
      6.9-1 Equalizer
      6.9-2 EqualizerOp
      6.9-3 EmbeddingOfEqualizer
      6.9-4 EmbeddingOfEqualizerOp
      6.9-5 EmbeddingOfEqualizerWithGivenEqualizer
      6.9-6 MorphismFromEqualizerToSink
      6.9-7 MorphismFromEqualizerToSinkOp
      6.9-8 MorphismFromEqualizerToSinkWithGivenEqualizer
      6.9-9 UniversalMorphismIntoEqualizer
      6.9-10 UniversalMorphismIntoEqualizerWithGivenEqualizer
      6.9-11 AddEqualizer
      6.9-12 AddEmbeddingOfEqualizer
      6.9-13 AddEmbeddingOfEqualizerWithGivenEqualizer
      6.9-14 AddMorphismFromEqualizerToSink
      6.9-15 AddMorphismFromEqualizerToSinkWithGivenEqualizer
      6.9-16 AddUniversalMorphismIntoEqualizer
      6.9-17 AddUniversalMorphismIntoEqualizerWithGivenEqualizer
      6.9-18 EqualizerFunctorial
      6.9-19 EqualizerFunctorialWithGivenEqualizers
      6.9-20 AddEqualizerFunctorialWithGivenEqualizers
    6.10 [33X[0;0YCoequalizer[133X
      6.10-1 Coequalizer
      6.10-2 CoequalizerOp
      6.10-3 ProjectionOntoCoequalizer
      6.10-4 ProjectionOntoCoequalizerOp
      6.10-5 ProjectionOntoCoequalizerWithGivenCoequalizer
      6.10-6 MorphismFromSourceToCoequalizer
      6.10-7 MorphismFromSourceToCoequalizerOp
      6.10-8 MorphismFromSourceToCoequalizerWithGivenCoequalizer
      6.10-9 UniversalMorphismFromCoequalizer
      6.10-10 UniversalMorphismFromCoequalizerWithGivenCoequalizer
      6.10-11 AddCoequalizer
      6.10-12 AddProjectionOntoCoequalizer
      6.10-13 AddProjectionOntoCoequalizerWithGivenCoequalizer
      6.10-14 AddMorphismFromSourceToCoequalizer
      6.10-15 AddMorphismFromSourceToCoequalizerWithGivenCoequalizer
      6.10-16 AddUniversalMorphismFromCoequalizer
      6.10-17 AddUniversalMorphismFromCoequalizerWithGivenCoequalizer
      6.10-18 CoequalizerFunctorial
      6.10-19 CoequalizerFunctorialWithGivenCoequalizers
      6.10-20 AddCoequalizerFunctorialWithGivenCoequalizers
    6.11 [33X[0;0YFiber Product[133X
      6.11-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference
      6.11-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp
      6.11-3 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference
      6.11-4 IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
      6.11-5 IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp
      6.11-6 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
      6.11-7 IsomorphismFromFiberProductToEqualizerOfDirectProductDiagram
      6.11-8 IsomorphismFromFiberProductToEqualizerOfDirectProductDiagramOp
      6.11-9 AddIsomorphismFromFiberProductToEqualizerOfDirectProductDiagram
      6.11-10 IsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct
      6.11-11 IsomorphismFromEqualizerOfDirectProductDiagramToFiberProductOp
      6.11-12 AddIsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct
      6.11-13 DirectSumDiagonalDifference
      6.11-14 DirectSumDiagonalDifferenceOp
      6.11-15 AddDirectSumDiagonalDifference
      6.11-16 FiberProductEmbeddingInDirectSum
      6.11-17 FiberProductEmbeddingInDirectSumOp
      6.11-18 AddFiberProductEmbeddingInDirectSum
      6.11-19 FiberProduct
      6.11-20 FiberProductOp
      6.11-21 ProjectionInFactorOfFiberProduct
      6.11-22 ProjectionInFactorOfFiberProductOp
      6.11-23 ProjectionInFactorOfFiberProductWithGivenFiberProduct
      6.11-24 MorphismFromFiberProductToSink
      6.11-25 MorphismFromFiberProductToSinkOp
      6.11-26 MorphismFromFiberProductToSinkWithGivenFiberProduct
      6.11-27 UniversalMorphismIntoFiberProduct
      6.11-28 UniversalMorphismIntoFiberProductOp
      6.11-29 UniversalMorphismIntoFiberProductWithGivenFiberProduct
      6.11-30 AddFiberProduct
      6.11-31 AddProjectionInFactorOfFiberProduct
      6.11-32 AddProjectionInFactorOfFiberProductWithGivenFiberProduct
      6.11-33 AddMorphismFromFiberProductToSink
      6.11-34 AddMorphismFromFiberProductToSinkWithGivenFiberProduct
      6.11-35 AddUniversalMorphismIntoFiberProduct
      6.11-36 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct
      6.11-37 FiberProductFunctorial
      6.11-38 FiberProductFunctorialWithGivenFiberProducts
      6.11-39 AddFiberProductFunctorialWithGivenFiberProducts
    6.12 [33X[0;0YPushout[133X
      6.12-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference
      6.12-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp
      6.12-3 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference
      6.12-4 IsomorphismFromCokernelOfDiagonalDifferenceToPushout
      6.12-5 IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp
      6.12-6 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout
      6.12-7 IsomorphismFromPushoutToCoequalizerOfCoproductDiagram
      6.12-8 IsomorphismFromPushoutToCoequalizerOfCoproductDiagramOp
      6.12-9 AddIsomorphismFromPushoutToCoequalizerOfCoproductDiagram
      6.12-10 IsomorphismFromCoequalizerOfCoproductDiagramToPushout
      6.12-11 IsomorphismFromCoequalizerOfCoproductDiagramToPushoutOp
      6.12-12 AddIsomorphismFromCoequalizerOfCoproductDiagramToPushout
      6.12-13 DirectSumCodiagonalDifference
      6.12-14 DirectSumCodiagonalDifferenceOp
      6.12-15 AddDirectSumCodiagonalDifference
      6.12-16 DirectSumProjectionInPushout
      6.12-17 DirectSumProjectionInPushoutOp
      6.12-18 AddDirectSumProjectionInPushout
      6.12-19 Pushout
      6.12-20 Pushout
      6.12-21 PushoutOp
      6.12-22 InjectionOfCofactorOfPushout
      6.12-23 InjectionOfCofactorOfPushoutOp
      6.12-24 InjectionOfCofactorOfPushoutWithGivenPushout
      6.12-25 MorphismFromSourceToPushout
      6.12-26 MorphismFromSourceToPushoutOp
      6.12-27 MorphismFromSourceToPushoutWithGivenPushout
      6.12-28 UniversalMorphismFromPushout
      6.12-29 UniversalMorphismFromPushoutOp
      6.12-30 UniversalMorphismFromPushoutWithGivenPushout
      6.12-31 AddPushout
      6.12-32 AddInjectionOfCofactorOfPushout
      6.12-33 AddInjectionOfCofactorOfPushoutWithGivenPushout
      6.12-34 AddMorphismFromSourceToPushout
      6.12-35 AddMorphismFromSourceToPushoutWithGivenPushout
      6.12-36 AddUniversalMorphismFromPushout
      6.12-37 AddUniversalMorphismFromPushoutWithGivenPushout
      6.12-38 PushoutFunctorial
      6.12-39 PushoutFunctorialWithGivenPushouts
      6.12-40 AddPushoutFunctorialWithGivenPushouts
    6.13 [33X[0;0YImage[133X
      6.13-1 IsomorphismFromImageObjectToKernelOfCokernel
      6.13-2 AddIsomorphismFromImageObjectToKernelOfCokernel
      6.13-3 IsomorphismFromKernelOfCokernelToImageObject
      6.13-4 AddIsomorphismFromKernelOfCokernelToImageObject
      6.13-5 ImageObject
      6.13-6 ImageEmbedding
      6.13-7 ImageEmbeddingWithGivenImageObject
      6.13-8 CoastrictionToImage
      6.13-9 CoastrictionToImageWithGivenImageObject
      6.13-10 UniversalMorphismFromImage
      6.13-11 UniversalMorphismFromImageWithGivenImageObject
      6.13-12 AddImageObject
      6.13-13 AddImageEmbedding
      6.13-14 AddImageEmbeddingWithGivenImageObject
      6.13-15 AddCoastrictionToImage
      6.13-16 AddCoastrictionToImageWithGivenImageObject
      6.13-17 AddUniversalMorphismFromImage
      6.13-18 AddUniversalMorphismFromImageWithGivenImageObject
    6.14 [33X[0;0YCoimage[133X
      6.14-1 MorphismFromCoimageToImage
      6.14-2 MorphismFromCoimageToImageWithGivenObjects
      6.14-3 AddMorphismFromCoimageToImageWithGivenObjects
      6.14-4 InverseMorphismFromCoimageToImage
      6.14-5 InverseMorphismFromCoimageToImageWithGivenObjects
      6.14-6 AddInverseMorphismFromCoimageToImageWithGivenObjects
      6.14-7 IsomorphismFromCoimageToCokernelOfKernel
      6.14-8 AddIsomorphismFromCoimageToCokernelOfKernel
      6.14-9 IsomorphismFromCokernelOfKernelToCoimage
      6.14-10 AddIsomorphismFromCokernelOfKernelToCoimage
      6.14-11 Coimage
      6.14-12 CoimageProjection
      6.14-13 CoimageProjectionWithGivenCoimage
      6.14-14 AstrictionToCoimage
      6.14-15 AstrictionToCoimageWithGivenCoimage
      6.14-16 UniversalMorphismIntoCoimage
      6.14-17 UniversalMorphismIntoCoimageWithGivenCoimage
      6.14-18 AddCoimage
      6.14-19 AddCoimageProjection
      6.14-20 AddCoimageProjectionWithGivenCoimage
      6.14-21 AddAstrictionToCoimage
      6.14-22 AddAstrictionToCoimageWithGivenCoimage
      6.14-23 AddUniversalMorphismIntoCoimage
      6.14-24 AddUniversalMorphismIntoCoimageWithGivenCoimage
      6.14-25 CanonicalIdentificationFromImageObjectToCoimage
      6.14-26 AddCanonicalIdentificationFromImageObjectToCoimage
      6.14-27 CanonicalIdentificationFromCoimageToImageObject
      6.14-28 AddCanonicalIdentificationFromCoimageToImageObject
    6.15 [33X[0;0YHomology objects[133X
      6.15-1 HomologyObject
      6.15-2 HomologyObjectFunctorial
      6.15-3 HomologyObjectFunctorialWithGivenHomologyObjects
      6.15-4 IsomorphismFromHomologyObjectToItsConstructionAsAnImageObject
      6.15-5 IsomorphismFromItsConstructionAsAnImageObjectToHomologyObject
      6.15-6 AddHomologyObject
      6.15-7 AddHomologyObjectFunctorialWithGivenHomologyObjects
  7 [33X[0;0YAdd Functions[133X
    7.1 [33X[0;0YFunctions Installed by Add[133X
    7.2 [33X[0;0YAdd Method[133X
    7.3 [33X[0;0YInstallAdd Function[133X
    7.4 [33X[0;0YEnhancing the method name record[133X
    7.5 [33X[0;0YInstall All Adds[133X
    7.6 [33X[0;0YPrepare functions[133X
      7.6-1 CAPOperationPrepareFunction
      7.6-2 CAPAddPrepareFunction
      7.6-3 ListCAPPrepareFunctions
  8 [33X[0;0YManaging Derived Methods[133X
    8.1 [33X[0;0YInfo Class[133X
      8.1-1 DerivationInfo
      8.1-2 ActivateDerivationInfo
      8.1-3 DeactivateDerivationInfo
    8.2 [33X[0;0YDerivation Objects[133X
      8.2-1 IsDerivedMethod
      8.2-2 MakeDerivation
      8.2-3 DerivationName
      8.2-4 DerivationWeight
      8.2-5 DerivationFunctionsWithExtraFilters
      8.2-6 CategoryFilter
      8.2-7 IsApplicableToCategory
      8.2-8 TargetOperation
      8.2-9 UsedOperations
      8.2-10 UsedOperationMultiples
      8.2-11 UsedOperationsWithMultiples
      8.2-12 InstallDerivationForCategory
      8.2-13 DerivationResultWeight
      8.2-14 FunctionCalledBeforeInstallation
    8.3 [33X[0;0YDerivation Graphs[133X
      8.3-1 IsDerivedMethodGraph
      8.3-2 MakeDerivationGraph
      8.3-3 AddOperationsToDerivationGraph
      8.3-4 AddDerivation
      8.3-5 AddDerivation
      8.3-6 AddDerivation
      8.3-7 AddDerivation
      8.3-8 AddDerivationPair
      8.3-9 AddDerivationPair
      8.3-10 AddDerivationPair
      8.3-11 AddDerivationPair
      8.3-12 AddDerivationToCAP
      8.3-13 AddDerivationPairToCAP
      8.3-14 AddWithGivenDerivationPairToCAP
      8.3-15 Operations
      8.3-16 DerivationsUsingOperation
      8.3-17 DerivationsOfOperation
    8.4 [33X[0;0YManaging Derivations in a Category[133X
      8.4-1 IsOperationWeightList
      8.4-2 MakeOperationWeightList
      8.4-3 DerivationGraph
      8.4-4 CategoryOfOperationWeightList
      8.4-5 CurrentOperationWeight
      8.4-6 OperationWeightUsingDerivation
      8.4-7 DerivationOfOperation
      8.4-8 InstallDerivationsUsingOperation
      8.4-9 Reevaluate
      8.4-10 Saturate
      8.4-11 AddPrimitiveOperation
      8.4-12 PrintDerivationTree
      8.4-13 PrintTree
      8.4-14 PrintTreeRec
    8.5 [33X[0;0YMin Heaps for Strings[133X
      8.5-1 IsStringMinHeap
      8.5-2 StringMinHeap
      8.5-3 Add
      8.5-4 ExtractMin
      8.5-5 DecreaseKey
      8.5-6 IsEmptyHeap
      8.5-7 HeapSize
      8.5-8 Contains
      8.5-9 Swap
      8.5-10 Heapify
  9 [33X[0;0YTechnical Details[133X
    9.1 [33X[0;0YThe Category Cat[133X
      9.1-1 ObjectCache
      9.1-2 MorphismCache
    9.2 [33X[0;0YInstall Functions for IsWellDefined[133X
      9.2-1 InstallMethodWithToDoForIsWellDefined
      9.2-2 InstallSetWithToDoForIsWellDefined
      9.2-3 DeclareAttributeWithToDoForIsWellDefined
      9.2-4 DeclareFamilyProperty
      9.2-5 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS
      9.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS
      9.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT
      9.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION
      9.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST
      9.2-10 CAP_INTERNAL_ASSERT_IS_CELL_OF_CATEGORY
      9.2-11 CAP_INTERNAL_ASSERT_IS_OBJECT_OF_CATEGORY
      9.2-12 CAP_INTERNAL_ASSERT_IS_MORPHISM_OF_CATEGORY
      9.2-13 CAP_INTERNAL_ASSERT_IS_TWO_CELL_OF_CATEGORY
      9.2-14 CachingStatistic
      9.2-15 BrowseCachingStatistic
  10 [33X[0;0YLimits and Colimits[133X
    10.1 [33X[0;0YSpecification of Limits and Colimits[133X
    10.2 [33X[0;0YEnhancing Limit Specifications[133X
    10.3 [33X[0;0YValidating entries of a method name record which are part of a limit
    or colimit[133X
  11 [33X[0;0YExamples and Tests[133X
    11.1 [33X[0;0YFunctors[133X
    11.2 [33X[0;0YHomomorphism structure[133X
    11.3 [33X[0;0YSpectral Sequences[133X
    11.4 [33X[0;0YHomology object[133X
    11.5 [33X[0;0YLiftable[133X
    11.6 [33X[0;0YMonoidal Categories[133X
    11.7 [33X[0;0YMorphismFromSourceToPushout and MorphismFromFiberProductToSink[133X
    11.8 [33X[0;0YOpposite category[133X
    11.9 [33X[0;0YGeneralized Morphisms Category[133X
    11.10 [33X[0;0YIsWellDefined[133X
    11.11 [33X[0;0YKernel[133X
    11.12 [33X[0;0YFiberProduct[133X
  
  
  [32X
