GAP (CAP) - Contents

  1.1-1 IsCapCategory
  1.1-2 IsCapCategoryCell
  1.1-3 IsCapCategoryObject
  1.1-4 IsCapCategoryMorphism
  1.1-5 IsCapCategoryTwoCell

1.2 Categorical properties

  1.2-1 AddCategoricalProperty
  1.2-2 IsObjectFiniteCategory
  1.2-3 IsFiniteCategory
  1.2-4 IsFinite
  1.2-5 IsEquivalentToFiniteCategory
  1.2-6 IsFinitelyPresentedCategory
  1.2-7 IsEquippedWithHomomorphismStructure
  1.2-8 IsCategoryWithDecidableLifts
  1.2-9 IsCategoryWithDecidableColifts
  1.2-10 IsCategoryWithInitialObject
  1.2-11 IsCategoryWithTerminalObject
  1.2-12 IsCategoryWithZeroObject
  1.2-13 IsCategoryWithEqualizers
  1.2-14 IsCategoryWithCoequalizers
  1.2-15 IsCategoryWithCoequalizersOfIdentityAndAutomorphisms
  1.2-16 IsCategoryWithEqualizersOfIdentityAndAutomorphisms
  1.2-17 IsEnrichedOverCommutativeRegularSemigroup
  1.2-18 IsSkeletalCategory
  1.2-19 IsAbCategory
  1.2-20 IsLinearCategoryOverCommutativeSemiring
  1.2-21 IsLinearCategoryOverCommutativeSemiringWithFinitelyGeneratedFreeExternalHoms
  1.2-22 IsLinearCategoryOverCommutativeRing
  1.2-23 IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
  1.2-24 IsCategoryWithKernels
  1.2-25 IsCategoryWithCokernels
  1.2-26 IsAdditiveCategory
  1.2-27 IsPreAbelianCategory
  1.2-28 IsAbelianCategory
  1.2-29 IsAbelianCategoryWithEnoughProjectives
  1.2-30 IsAbelianCategoryWithEnoughInjectives
  1.2-31 IsLocallyOfFiniteProjectiveDimension
  1.2-32 IsLocallyOfFiniteInjectiveDimension
  1.2-33 ListOfDefiningOperations

1.3 Constructor

  1.3-1 CreateCapCategory
  1.3-2 CreateCapCategory
  1.3-3 CreateCapCategory
  1.3-4 CreateCapCategoryWithDataTypes

1.4 Internal Attributes

  1.4-1 Name
  1.4-2 CategoryFilter
  1.4-3 ObjectFilter
  1.4-4 MorphismFilter
  1.4-5 TwoCellFilter
  1.4-6 ObjectDatumType
  1.4-7 MorphismDatumType
  1.4-8 TwoCellDatumType
  1.4-9 CommutativeSemiringOfLinearCategory
  1.4-10 RangeCategoryOfHomomorphismStructure
  1.4-11 AdditiveGenerators
  1.4-12 IndecomposableProjectiveObjects
  1.4-13 IndecomposableInjectiveObjects

1.5 Logic switcher

  1.5-1 CapCategorySwitchLogicPropagationForObjectsOn
  1.5-2 CapCategorySwitchLogicPropagationForObjectsOff
  1.5-3 CapCategorySwitchLogicPropagationForMorphismsOn
  1.5-4 CapCategorySwitchLogicPropagationForMorphismsOff
  1.5-5 CapCategorySwitchLogicPropagationOn
  1.5-6 CapCategorySwitchLogicPropagationOff
  1.5-7 CapCategorySwitchLogicOn
  1.5-8 CapCategorySwitchLogicOff

1.6 Tool functions

  1.6-1 CanCompute
  1.6-2 OperationWeight
  1.6-3 MissingOperationsForConstructivenessOfCategory

1.7 Well-Definedness of Cells

1.7-1 IsWellDefined

1.8 Unpacking data structures

  1.8-1 Down
  1.8-2 DownOnlyMorphismData
  1.8-3 DownToBottom

1.9 Caching

  1.9-1 SetCachingOfCategory
  1.9-2 SetCachingOfCategoryWeak
  1.9-3 SetDefaultCaching

1.10 Sanity checks

1.10-1 DisableInputSanityChecks

1.11 Timing statistics

1.11-1 EnableTimingStatistics

1.12 Enable automatic calls of Add

1.12-1 EnableAddForCategoricalOperations

1.13 Performance tweaks

1.14 LaTeX

1.14-1 LaTeXOutput
1.14-2 LaTeXOutput

2 Objects

2.1 Attributes for the Type of Objects

2.1-1 CapCategory

2.2 Adding Objects to a Category

  2.2-1 Add
  2.2-2 AddObject
  2.2-3 CreateCapCategoryObjectWithAttributes
  2.2-4 AsCapCategoryObject
  2.2-5 AsPrimitiveValue

2.3 Equalities for Objects

  2.3-1 IsEqualForObjects
  2.3-2 IsIsomorphicForObjects
  2.3-3 SomeIsomorphismBetweenObjects

2.4 Categorical Properties of Objects

  2.4-1 IsBijectiveObject
  2.4-2 IsProjective
  2.4-3 IsInjective
  2.4-4 IsTerminal
  2.4-5 IsInitial
  2.4-6 IsZeroForObjects
  2.4-7 IsZero

2.5 Random Objects

  2.5-1 RandomObjectByInteger
  2.5-2 RandomObjectByList
  2.5-3 RandomObject
  2.5-4 RandomObject

2.6 Tool functions for caches

2.6-1 IsEqualForCacheForObjects

2.7 Object constructors

  2.7-1 ObjectConstructor
  2.7-2 /
  2.7-3 ObjectDatum

2.8 Well-Definedness of Objects

2.8-1 IsWellDefinedForObjects

2.9 SetOfObjects

2.9-1 SetOfObjectsOfCategory
2.9-2 SetOfObjects

2.10 Projectives

  2.10-1 SomeProjectiveObject
  2.10-2 EpimorphismFromSomeProjectiveObject
  2.10-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
  2.10-4 ProjectiveLift

2.11 Injectives

  2.11-1 SomeInjectiveObject
  2.11-2 MonomorphismIntoSomeInjectiveObject
  2.11-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
  2.11-4 InjectiveColift

2.12 Simplified Objects

  2.12-1 Simplify
  2.12-2 SimplifyObject
  2.12-3 SimplifyObject_IsoFromInputObject
  2.12-4 SimplifyObject_IsoToInputObject

2.13 Dimensions

2.13-1 ProjectiveDimension
2.13-2 InjectiveDimension

3 Morphisms

3.1 Attributes for the Type of Morphisms

  3.1-1 CapCategory
  3.1-2 Source
  3.1-3 Range
  3.1-4 Target

3.2 Adding Morphisms to a Category

  3.2-1 Add
  3.2-2 AddMorphism
  3.2-3 CreateCapCategoryMorphismWithAttributes
  3.2-4 AsCapCategoryMorphism
  3.2-5 AsPrimitiveValue

3.3 Morphism constructors

3.3-1 MorphismConstructor
3.3-2 MorphismDatum

3.4 Categorical Properties of Morphisms

  3.4-1 IsMonomorphism
  3.4-2 IsEpimorphism
  3.4-3 IsIsomorphism
  3.4-4 IsSplitMonomorphism
  3.4-5 IsSplitEpimorphism
  3.4-6 IsOne
  3.4-7 IsIdempotent

3.5 Random Morphisms

  3.5-1 RandomMorphismWithFixedSourceByInteger
  3.5-2 RandomMorphismWithFixedSourceByList
  3.5-3 RandomMorphismWithFixedRangeByInteger
  3.5-4 RandomMorphismWithFixedRangeByList
  3.5-5 RandomMorphismWithFixedSourceAndRangeByInteger
  3.5-6 RandomMorphismWithFixedSourceAndRangeByList
  3.5-7 RandomMorphismByInteger
  3.5-8 RandomMorphismByList
  3.5-9 RandomMorphismWithFixedSource

3.6 Non-Categorical Properties of Morphisms

  3.6-1 IsEqualToIdentityMorphism
  3.6-2 IsEqualToZeroMorphism
  3.6-3 IsEndomorphism
  3.6-4 IsAutomorphism

3.7 Equality and Congruence for Morphisms

  3.7-1 IsCongruentForMorphisms
  3.7-2 IsEqualForMorphisms
  3.7-3 IsEqualForMorphismsOnMor

3.8 Basic Operations for Morphisms in Ab-Categories

  3.8-1 IsZeroForMorphisms
  3.8-2 AdditionForMorphisms
  3.8-3 SubtractionForMorphisms
  3.8-4 AdditiveInverseForMorphisms
  3.8-5 MultiplyWithElementOfCommutativeSemiringForMorphisms
  3.8-6 *
  3.8-7 ZeroMorphism

3.9 Subobject and Factorobject Operations

  3.9-1 IsEqualAsSubobjects
  3.9-2 IsEqualAsFactorobjects
  3.9-3 IsDominating
  3.9-4 IsCodominating

3.10 Identity Morphism and Composition of Morphisms

  3.10-1 IdentityMorphism
  3.10-2 PreCompose
  3.10-3 PreCompose
  3.10-4 PreComposeList
  3.10-5 PostCompose
  3.10-6 PostCompose
  3.10-7 PostComposeList
  3.10-8 SumOfMorphisms
  3.10-9 LinearCombinationOfMorphisms

3.11 Well-Definedness of Morphisms

3.11-1 IsWellDefinedForMorphisms
3.11-2 IsWellDefinedForMorphismsWithGivenSourceAndRange

3.12 Lift/Colift

  3.12-1 LiftAlongMonomorphism
  3.12-2 ColiftAlongEpimorphism
  3.12-3 IsLiftableAlongMonomorphism
  3.12-4 IsColiftableAlongEpimorphism
  3.12-5 Lift
  3.12-6 LiftOrFail
  3.12-7 IsLiftable
  3.12-8 Colift
  3.12-9 ColiftOrFail
  3.12-10 IsColiftable

3.13 Inverses

  3.13-1 InverseForMorphisms
  3.13-2 PreInverseForMorphisms
  3.13-3 PostInverseForMorphisms

3.14 Tool functions for caches

3.14-1 IsEqualForCacheForMorphisms

3.15 IsHomSetInhabited

3.15-1 IsHomSetInhabited

3.16 SetOfMorphisms

3.16-1 SetOfMorphismsOfFiniteCategory
3.16-2 SetOfMorphisms

3.17 SetOfGeneratingMorphisms

3.17-1 SetOfGeneratingMorphismsOfCategory
3.17-2 SetOfGeneratingMorphisms

3.18 Homomorphism structures

3.19 Simplified Morphisms

  3.19-1 SimplifyMorphism
  3.19-2 SimplifySource
  3.19-3 SimplifySource_IsoToInputObject
  3.19-4 SimplifySource_IsoFromInputObject
  3.19-5 SimplifyRange
  3.19-6 SimplifyRange_IsoToInputObject
  3.19-7 SimplifyRange_IsoFromInputObject
  3.19-8 SimplifySourceAndRange
  3.19-9 SimplifySourceAndRange_IsoToInputRange
  3.19-10 SimplifySourceAndRange_IsoFromInputRange
  3.19-11 SimplifySourceAndRange_IsoToInputSource
  3.19-12 SimplifySourceAndRange_IsoFromInputSource
  3.19-13 SimplifyEndo
  3.19-14 SimplifyEndo_IsoToInputObject
  3.19-15 SimplifyEndo_IsoFromInputObject
  3.19-16 Simplify

3.20 Reduction by split epi summands

  3.20-1 SomeReductionBySplitEpiSummand
  3.20-2 SomeReductionBySplitEpiSummand_MorphismToInputRange
  3.20-3 SomeReductionBySplitEpiSummand_MorphismFromInputRange

4 Category 2-Cells

4.1 Attributes for the Type of 2-Cells

  4.1-1 Source
  4.1-2 Range
  4.1-3 Target

4.2 Adding 2-Cells to a Category

  4.2-1 Add
  4.2-2 AddTwoCell
  4.2-3 CreateCapCategoryTwoCellWithAttributes

4.3 Identity 2-Cell and Composition of 2-Cells

  4.3-1 IdentityTwoCell
  4.3-2 HorizontalPreCompose
  4.3-3 HorizontalPostCompose
  4.3-4 VerticalPreCompose
  4.3-5 VerticalPostCompose

4.4 Well-Definedness for 2-Cells

4.4-1 IsWellDefinedForTwoCells

5 Category of Categories

5.1 The Category Cat

5.1-1 CapCat

5.2 Categories

  5.2-1 IsCapCategoryAsCatObject
  5.2-2 IsCapFunctor
  5.2-3 IsCapNaturalTransformation

5.3 Constructors

5.3-1 AsCatObject
5.3-2 AsCapCategory

5.4 Functors

  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 Natural transformations

  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 Universal Objects

6.1 Kernel

  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 KernelObjectFunctorial
  6.1-9 KernelObjectFunctorial
  6.1-10 KernelObjectFunctorialWithGivenKernelObjects
  6.1-11 KernelObjectFunctorialWithGivenKernelObjects

6.2 Cokernel

  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 CokernelObjectFunctorial
  6.2-9 CokernelObjectFunctorial
  6.2-10 CokernelObjectFunctorialWithGivenCokernelObjects
  6.2-11 CokernelObjectFunctorialWithGivenCokernelObjects

6.3 Zero Object

  6.3-1 ZeroObject
  6.3-2 ZeroObject
  6.3-3 UniversalMorphismFromZeroObject
  6.3-4 UniversalMorphismFromZeroObjectWithGivenZeroObject
  6.3-5 UniversalMorphismIntoZeroObject
  6.3-6 UniversalMorphismIntoZeroObjectWithGivenZeroObject
  6.3-7 MorphismFromZeroObject
  6.3-8 MorphismIntoZeroObject
  6.3-9 IsomorphismFromZeroObjectToInitialObject
  6.3-10 IsomorphismFromInitialObjectToZeroObject
  6.3-11 IsomorphismFromZeroObjectToTerminalObject
  6.3-12 IsomorphismFromTerminalObjectToZeroObject
  6.3-13 ZeroObjectFunctorial
  6.3-14 ZeroObjectFunctorialWithGivenZeroObjects

6.4 Terminal Object

  6.4-1 TerminalObject
  6.4-2 TerminalObject
  6.4-3 UniversalMorphismIntoTerminalObject
  6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject
  6.4-5 TerminalObjectFunctorial
  6.4-6 TerminalObjectFunctorialWithGivenTerminalObjects

6.5 Initial Object

  6.5-1 InitialObject
  6.5-2 InitialObject
  6.5-3 UniversalMorphismFromInitialObject
  6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject
  6.5-5 InitialObjectFunctorial
  6.5-6 InitialObjectFunctorialWithGivenInitialObjects

6.6 Direct Sum

  6.6-1 DirectSum
  6.6-2 DirectSumOp
  6.6-3 ProjectionInFactorOfDirectSum
  6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum
  6.6-5 InjectionOfCofactorOfDirectSum
  6.6-6 InjectionOfCofactorOfDirectSumWithGivenDirectSum
  6.6-7 UniversalMorphismIntoDirectSum
  6.6-8 UniversalMorphismIntoDirectSumWithGivenDirectSum
  6.6-9 UniversalMorphismFromDirectSum
  6.6-10 UniversalMorphismFromDirectSumWithGivenDirectSum
  6.6-11 IsomorphismFromDirectSumToDirectProduct
  6.6-12 IsomorphismFromDirectProductToDirectSum
  6.6-13 IsomorphismFromDirectSumToCoproduct
  6.6-14 IsomorphismFromCoproductToDirectSum
  6.6-15 MorphismBetweenDirectSums
  6.6-16 MorphismBetweenDirectSums
  6.6-17 MorphismBetweenDirectSumsWithGivenDirectSums
  6.6-18 ComponentOfMorphismIntoDirectSum
  6.6-19 ComponentOfMorphismFromDirectSum
  6.6-20 DirectSumFunctorial
  6.6-21 DirectSumFunctorialWithGivenDirectSums

6.7 Coproduct

  6.7-1 Coproduct
  6.7-2 Coproduct
  6.7-3 Coproduct
  6.7-4 InjectionOfCofactorOfCoproduct
  6.7-5 InjectionOfCofactorOfCoproductWithGivenCoproduct
  6.7-6 UniversalMorphismFromCoproduct
  6.7-7 UniversalMorphismFromCoproductWithGivenCoproduct
  6.7-8 CoproductFunctorial
  6.7-9 CoproductFunctorialWithGivenCoproducts
  6.7-10 ComponentOfMorphismFromCoproduct

6.8 Direct Product

  6.8-1 DirectProduct
  6.8-2 DirectProductOp
  6.8-3 ProjectionInFactorOfDirectProduct
  6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct
  6.8-5 UniversalMorphismIntoDirectProduct
  6.8-6 UniversalMorphismIntoDirectProductWithGivenDirectProduct
  6.8-7 DirectProductFunctorial
  6.8-8 DirectProductFunctorialWithGivenDirectProducts
  6.8-9 ComponentOfMorphismIntoDirectProduct

6.9 Equalizer

  6.9-1 Equalizer
  6.9-2 EqualizerOp
  6.9-3 EmbeddingOfEqualizer
  6.9-4 EmbeddingOfEqualizerWithGivenEqualizer
  6.9-5 MorphismFromEqualizerToSink
  6.9-6 MorphismFromEqualizerToSinkWithGivenEqualizer
  6.9-7 UniversalMorphismIntoEqualizer
  6.9-8 UniversalMorphismIntoEqualizerWithGivenEqualizer
  6.9-9 EqualizerFunctorial
  6.9-10 EqualizerFunctorialWithGivenEqualizers
  6.9-11 JointPairwiseDifferencesOfMorphismsIntoDirectProduct
  6.9-12 IsomorphismFromEqualizerToKernelOfJointPairwiseDifferencesOfMorphismsIntoDirectProduct
  6.9-13 IsomorphismFromKernelOfJointPairwiseDifferencesOfMorphismsIntoDirectProductToEqualizer

6.10 Coequalizer

  6.10-1 Coequalizer
  6.10-2 CoequalizerOp
  6.10-3 ProjectionOntoCoequalizer
  6.10-4 ProjectionOntoCoequalizerWithGivenCoequalizer
  6.10-5 MorphismFromSourceToCoequalizer
  6.10-6 MorphismFromSourceToCoequalizerWithGivenCoequalizer
  6.10-7 UniversalMorphismFromCoequalizer
  6.10-8 UniversalMorphismFromCoequalizerWithGivenCoequalizer
  6.10-9 CoequalizerFunctorial
  6.10-10 CoequalizerFunctorialWithGivenCoequalizers
  6.10-11 JointPairwiseDifferencesOfMorphismsFromCoproduct
  6.10-12 IsomorphismFromCoequalizerToCokernelOfJointPairwiseDifferencesOfMorphismsFromCoproduct
  6.10-13 IsomorphismFromCokernelOfJointPairwiseDifferencesOfMorphismsFromCoproductToCoequalizer

6.11 CoequalizerOfIdentityAndAutomorphisms

6.12 Fiber Product (= Pullback)

  6.12-1 IsomorphismFromFiberProductToEqualizerOfDirectProductDiagram
  6.12-2 IsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct
  6.12-3 FiberProductEmbeddingInDirectProduct
  6.12-4 FiberProductEmbeddingInDirectSum
  6.12-5 FiberProduct
  6.12-6 FiberProductOp
  6.12-7 ProjectionInFactorOfFiberProduct
  6.12-8 ProjectionInFactorOfFiberProductWithGivenFiberProduct
  6.12-9 MorphismFromFiberProductToSink
  6.12-10 MorphismFromFiberProductToSinkWithGivenFiberProduct
  6.12-11 UniversalMorphismIntoFiberProduct
  6.12-12 UniversalMorphismIntoFiberProductWithGivenFiberProduct
  6.12-13 FiberProductFunctorial
  6.12-14 FiberProductFunctorialWithGivenFiberProducts

6.13 Pushout

  6.13-1 IsomorphismFromPushoutToCoequalizerOfCoproductDiagram
  6.13-2 IsomorphismFromCoequalizerOfCoproductDiagramToPushout
  6.13-3 PushoutProjectionFromCoproduct
  6.13-4 PushoutProjectionFromDirectSum
  6.13-5 Pushout
  6.13-6 Pushout
  6.13-7 InjectionOfCofactorOfPushout
  6.13-8 InjectionOfCofactorOfPushoutWithGivenPushout
  6.13-9 MorphismFromSourceToPushout
  6.13-10 MorphismFromSourceToPushoutWithGivenPushout
  6.13-11 UniversalMorphismFromPushout
  6.13-12 UniversalMorphismFromPushoutWithGivenPushout
  6.13-13 PushoutFunctorial
  6.13-14 PushoutFunctorialWithGivenPushouts

6.14 Image

  6.14-1 IsomorphismFromImageObjectToKernelOfCokernel
  6.14-2 IsomorphismFromKernelOfCokernelToImageObject
  6.14-3 ImageObject
  6.14-4 ImageEmbedding
  6.14-5 ImageEmbeddingWithGivenImageObject
  6.14-6 CoastrictionToImage
  6.14-7 CoastrictionToImageWithGivenImageObject
  6.14-8 UniversalMorphismFromImage
  6.14-9 UniversalMorphismFromImageWithGivenImageObject
  6.14-10 ImageObjectFunctorial
  6.14-11 ImageObjectFunctorialWithGivenImageObjects

6.15 Coimage

  6.15-1 IsomorphismFromCoimageToCokernelOfKernel
  6.15-2 IsomorphismFromCokernelOfKernelToCoimage
  6.15-3 CoimageObject
  6.15-4 CoimageProjection
  6.15-5 CoimageProjectionWithGivenCoimageObject
  6.15-6 AstrictionToCoimage
  6.15-7 AstrictionToCoimageWithGivenCoimageObject
  6.15-8 UniversalMorphismIntoCoimage
  6.15-9 UniversalMorphismIntoCoimageWithGivenCoimageObject
  6.15-10 CoimageObjectFunctorial
  6.15-11 CoimageObjectFunctorialWithGivenCoimageObjects

6.16 Morphism between Coimage and Image

  6.16-1 MorphismFromCoimageToImage
  6.16-2 MorphismFromCoimageToImageWithGivenObjects
  6.16-3 InverseOfMorphismFromCoimageToImage
  6.16-4 InverseOfMorphismFromCoimageToImageWithGivenObjects

6.17 Homology objects

  6.17-1 HomologyObject
  6.17-2 HomologyObjectFunctorial
  6.17-3 HomologyObjectFunctorialWithGivenHomologyObjects
  6.17-4 IsomorphismFromHomologyObjectToItsConstructionAsAnImageObject
  6.17-5 IsomorphismFromItsConstructionAsAnImageObjectToHomologyObject

6.18 Projective covers and injective envelopes

  6.18-1 ProjectiveCoverObject
  6.18-2 EpimorphismFromProjectiveCoverObject
  6.18-3 EpimorphismFromProjectiveCoverObjectWithGivenProjectiveCoverObject
  6.18-4 InjectiveEnvelopeObject
  6.18-5 MonomorphismIntoInjectiveEnvelopeObject
  6.18-6 MonomorphismIntoInjectiveEnvelopeObjectWithGivenInjectiveEnvelopeObject

7 Add Functions

7.1 Functions Installed by Add

7.2 Add Method

7.3 Method name record entries

7.4 Enhancing the method name record

7.5 Prepare functions

  7.5-1 CAPOperationPrepareFunction
  7.5-2 CAPAddPrepareFunction
  7.5-3 ListCAPPrepareFunctions

7.6 Available Add functions

8 Managing Derived Methods

8.1 Info Class

  8.1-1 DerivationInfo
  8.1-2 ActivateDerivationInfo
  8.1-3 DeactivateDerivationInfo

8.2 Derivation Objects

  8.2-1 IsDerivedMethod
  8.2-2 CreateDerivation
  8.2-3 Description
  8.2-4 AdditionalWeight
  8.2-5 DerivationFunction
  8.2-6 CategoryFilter
  8.2-7 IsApplicableToCategory
  8.2-8 TargetOperation
  8.2-9 UsedOperationsWithMultiplesAndCategoryGetters
  8.2-10 InstallDerivationForCategory
  8.2-11 FunctionCalledBeforeInstallation

8.3 Derivation Graphs

  8.3-1 IsDerivedMethodGraph
  8.3-2 MakeDerivationGraph
  8.3-3 AddOperationsToDerivationGraph
  8.3-4 AddDerivation
  8.3-5 AddDerivationToCAP
  8.3-6 Operations
  8.3-7 DerivationsUsingOperation
  8.3-8 DerivationsOfOperation

8.4 Managing Derivations in a Category

  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 TriggerDerivationsUsingOperation
  8.4-9 Reevaluate
  8.4-10 Saturate
  8.4-11 AddPrimitiveOperation
  8.4-12 PrintDerivationTree
  8.4-13 PrintTree

9 Technical Details

9.1 The Category Cat

9.1-1 ObjectCache
9.1-2 MorphismCache

9.2 Tools

  9.2-1 FunctionWithNamedArguments
  9.2-2 CAP_INTERNAL_GET_DATA_TYPE_FROM_STRING
  9.2-3 CAP_INTERNAL_GET_DATA_TYPES_FROM_STRINGS
  9.2-4 CAP_INTERNAL_REPLACED_STRING_WITH_FILTER
  9.2-5 CAP_INTERNAL_REPLACED_STRINGS_WITH_FILTERS
  9.2-6 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT
  9.2-7 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION
  9.2-8 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST
  9.2-9 CAP_INTERNAL_ASSERT_VALUE_IS_OF_TYPE_GETTER
  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
  9.2-16 InstallDeprecatedAlias
  9.2-17 IsSpecializationOfFilter
  9.2-18 IsSpecializationOfFilterList
  9.2-19 InstallMethodForCompilerForCAP
  9.2-20 InstallOtherMethodForCompilerForCAP
  9.2-21 CapJitAddKnownMethod
  9.2-22 CapJitAddTypeSignature
  9.2-23 CapJitDataTypeOfListOf
  9.2-24 CapJitDataTypeOfNTupleOf
  9.2-25 CapJitDataTypeOfGroup
  9.2-26 CapJitDataTypeOfRing
  9.2-27 CapJitDataTypeOfCategory
  9.2-28 CapJitTypedExpression
  9.2-29 CapFixpoint
  9.2-30 Iterated
  9.2-31 Iterated
  9.2-32 TransitivelyNeededOtherPackages
  9.2-33 PackageOfCAPOperation
  9.2-34 SafePosition
  9.2-35 SafeUniquePosition
  9.2-36 SafePositionProperty
  9.2-37 SafeUniquePositionProperty
  9.2-38 SafeFirst
  9.2-39 SafeUniqueEntry
  9.2-40 NTuple
  9.2-41 Pair
  9.2-42 Triple
  9.2-43 TransposedMatWithGivenDimensions
  9.2-44 HandlePrecompiledTowers
  9.2-45 CAP_JIT_INCOMPLETE_LOGIC
  9.2-46 CAP_JIT_EXPR_CASE_WRAPPER
  9.2-47 ListWithKeys
  9.2-48 SumWithKeys
  9.2-49 ProductWithKeys
  9.2-50 ForAllWithKeys
  9.2-51 ForAnyWithKeys
  9.2-52 NumberWithKeys
  9.2-53 FilteredWithKeys
  9.2-54 FirstWithKeys
  9.2-55 LastWithKeys
  9.2-56 CreateGapObjectWithAttributes

10 Limits and Colimits

10.1 Specification of Limits and Colimits

10.2 Enhancing Limit Specifications

10.3 Functions

10.3-1 CAP_INTERNAL_GENERATE_CONVENIENCE_METHODS_FOR_LIMITS
10.3-2 CAP_INTERNAL_VALIDATE_LIMITS_IN_NAME_RECORD

11 The Category Constructor

11.1 Info class

11.1-1 InfoCategoryConstructor

11.2 Constructors

11.2-1 CategoryConstructor

12 Reinterpretations of categories

12.1 Introduction

12.2 Tutorial

12.3 Implementation details

12.4 Relation to CompilerForCAP

12.5 Attributes

12.5-1 ModelingCategory

12.6 Constructors

  12.6-1 ReinterpretationOfCategory
  12.6-2 ReinterpretationFunctor
  12.6-3 ModelingObject
  12.6-4 ReinterpretationOfObject
  12.6-5 ModelingMorphism
  12.6-6 ReinterpretationOfMorphism

13 Create wrapper hulls of a category

13.1 GAP categories

  13.1-1 IsWrapperCapCategory
  13.1-2 IsWrapperCapCategoryObject
  13.1-3 IsWrapperCapCategoryMorphism

13.2 Attributes

13.2-1 UnderlyingCell
13.2-2 UnderlyingCell

13.3 Constructors

  13.3-1 AsObjectInWrapperCategory
  13.3-2 AsMorphismInWrapperCategory
  13.3-3 AsMorphismInWrapperCategory
  13.3-4 /
  13.3-5 WrapperCategory
  13.3-6 WrappingFunctor

14 Dummy implementations

14.1 Dummy rings

  14.1-1 IsDummySemiring
  14.1-2 IsDummySemiringElement
  14.1-3 IsDummyCommutativeSemiring
  14.1-4 IsDummyCommutativeSemiringElement
  14.1-5 IsDummyRing
  14.1-6 IsDummyRingElement
  14.1-7 IsDummyCommutativeRing
  14.1-8 IsDummyCommutativeRingElement
  14.1-9 IsDummyField
  14.1-10 IsDummyFieldElement
  14.1-11 DummySemiring
  14.1-12 DummyCommutativeSemiring
  14.1-13 DummyRing
  14.1-14 DummyCommutativeRing
  14.1-15 DummyField

14.2 Dummy categories

  14.2-1 IsDummyCategory
  14.2-2 IsDummyCategoryObject
  14.2-3 IsDummyCategoryMorphism
  14.2-4 DummyCategory

14.3 Dummy homalg rings

  14.3-1 IsDummyHomalgRing
  14.3-2 IsDummyHomalgRingElement
  14.3-3 IsDummyCommutativeHomalgRing
  14.3-4 IsDummyCommutativeHomalgRingElement
  14.3-5 IsDummyHomalgField
  14.3-6 IsDummyHomalgFieldElement
  14.3-7 DummyHomalgRing
  14.3-8 DummyCommutativeHomalgRing
  14.3-9 DummyHomalgField

15 Terminal category

15.1 GAP Categories

  15.1-1 IsCapTerminalCategoryWithSingleObject
  15.1-2 IsObjectInCapTerminalCategoryWithSingleObject
  15.1-3 IsMorphismInCapTerminalCategoryWithSingleObject
  15.1-4 IsCapTerminalCategoryWithMultipleObjects
  15.1-5 IsObjectInCapTerminalCategoryWithMultipleObjects
  15.1-6 IsMorphismInCapTerminalCategoryWithMultipleObjects
  15.1-7 IsTerminalCategory

15.2 Constructors

  15.2-1 TerminalCategoryWithSingleObject
  15.2-2 TerminalCategoryWithMultipleObjects
  15.2-3 CAP_INTERNAL_CONSTRUCTOR_FOR_TERMINAL_CATEGORY

15.3 Attributes

  15.3-1 TerminalCategoryWithSingleObjectUniqueObject
  15.3-2 TerminalCategoryWithSingleObjectUniqueMorphism
  15.3-3 FunctorFromTerminalCategory

16 Finite skeletal discrete categories

16.1 GAP Categories

  16.1-1 IsFiniteSkeletalDiscreteCategory
  16.1-2 IsObjectInFiniteSkeletalDiscreteCategory
  16.1-3 IsMorphismInFiniteSkeletalDiscreteCategory

16.2 Constructors

16.2-1 FiniteSkeletalDiscreteCategory

16.3 Attributes

16.3-1 UnderlyingListOfGapObjects
16.3-2 UnderlyingGapObject

16.4 Properties

16.4-1 IsDiscreteCategory

17 Examples and Tests

17.1 Dummy implementations

  17.1-1 Dummy categories
  17.1-2 Dummy rings
  17.1-3 Dummy semirings

17.2 Finite skeletal discrete categories

17.3 Functors

17.4 HandlePrecompiledTowers

17.5 Terminal category

Index

CAP

Categories, Algorithms, Programming

Contents