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CAP

Categories, Algorithms, Programming

2017.07.25

25 July 2017

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: http://www.uni-siegen.de/fb6/rmi/
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

Sebastian Posur
Email: sebastian.posur@uni-siegen.de
Homepage: http://www.uni-siegen.de/fb6/rmi/
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

Øystein Skartsæterhagen
Email: oysteini@math.ntnu.no
Homepage: http://www.math.ntnu.no/~oysteini/
Address:
NTNU
Institutt for matematiske fag
7491 Trondheim
Norway

Contents

1 CAP Categories
 1.1 Categories

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

  1.2-1 CreateCapCategory
  1.2-2 CreateCapCategory
 1.3 Internal Attributes

  1.3-1 CategoryFilter
  1.3-2 CellFilter
  1.3-3 ObjectFilter
  1.3-4 MorphismFilter
  1.3-5 TwoCellFilter
 1.4 Logic switcher

  1.4-1 CapCategorySwitchLogicOn
  1.4-2 CapCategorySwitchLogicOff
 1.5 Tool functions

  1.5-1 CanCompute
  1.5-2 CheckConstructivenessOfCategory
 1.6 Well-Definedness of Cells

  1.6-1 IsWellDefined
 1.7 Type check

  1.7-1 DisableBasicOperationTypeCheck
2 Category of Categories
 2.1 The Category Cat

  2.1-1 CapCat
 2.2 Categories

  2.2-1 IsCapCategoryAsCatObject
  2.2-2 IsCapFunctor
  2.2-3 IsCapNaturalTransformation
 2.3 Constructors

  2.3-1 AsCatObject
  2.3-2 AsCapCategory
 2.4 Functors

  2.4-1 CapFunctor
  2.4-2 AddObjectFunction
  2.4-3 FunctorObjectOperation
  2.4-4 AddMorphismFunction
  2.4-5 FunctorMorphismOperation
  2.4-6 ApplyFunctor
  2.4-7 InstallFunctor
  2.4-8 IdentityFunctor
  2.4-9 FunctorCanonicalizeZeroObjects
  2.4-10 NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects
  2.4-11 FunctorCanonicalizeZeroMorphisms
  2.4-12 NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms
 2.5 Natural transformations

  2.5-1 Name
  2.5-2 NaturalTransformation
  2.5-3 AddNaturalTransformationFunction
  2.5-4 ApplyNaturalTransformation
  2.5-5 InstallNaturalTransformation
  2.5-6 HorizontalPreComposeNaturalTransformationWithFunctor
  2.5-7 HorizontalPreComposeFunctorWithNaturalTransformation
3 Morphisms
 3.1 Attributes for the Type of Morphisms

  3.1-1 CapCategory
  3.1-2 Source
  3.1-3 Range
 3.2 Categorical Properties of Morphisms

  3.2-1 AddIsMonomorphism
  3.2-2 AddIsEpimorphism
  3.2-3 AddIsIsomorphism
  3.2-4 AddIsSplitMonomorphism
  3.2-5 AddIsSplitEpimorphism
  3.2-6 AddIsOne
  3.2-7 AddIsIdempotent
 3.3 Non-Categorical Properties of Morphisms

  3.3-1 IsIdenticalToIdentityMorphism
  3.3-2 AddIsIdenticalToIdentityMorphism
  3.3-3 IsIdenticalToZeroMorphism
  3.3-4 AddIsIdenticalToZeroMorphism
  3.3-5 AddIsEndomorphism
  3.3-6 AddIsAutomorphism
 3.4 Equality and Congruence for Morphisms

  3.4-1 IsCongruentForMorphisms
  3.4-2 AddIsCongruentForMorphisms
  3.4-3 IsEqualForMorphisms
  3.4-4 AddIsEqualForMorphisms
  3.4-5 IsEqualForMorphismsOnMor
  3.4-6 AddIsEqualForMorphismsOnMor
 3.5 Basic Operations for Morphisms in Ab-Categories

  3.5-1 IsZeroForMorphisms
  3.5-2 AddIsZeroForMorphisms
  3.5-3 AdditionForMorphisms
  3.5-4 AddAdditionForMorphisms
  3.5-5 AdditiveInverseForMorphisms
  3.5-6 AddAdditiveInverseForMorphisms
  3.5-7 ZeroMorphism
  3.5-8 AddZeroMorphism
 3.6 Subobject and Factorobject Operations

  3.6-1 IsEqualAsSubobjects
  3.6-2 AddIsEqualAsSubobjects
  3.6-3 IsEqualAsFactorobjects
  3.6-4 AddIsEqualAsFactorobjects
  3.6-5 IsDominating
  3.6-6 AddIsDominating
  3.6-7 IsCodominating
  3.6-8 AddIsCodominating
 3.7 Identity Morphism and Composition of Morphisms

  3.7-1 IdentityMorphism
  3.7-2 AddIdentityMorphism
  3.7-3 PreCompose
  3.7-4 PreCompose
  3.7-5 AddPreCompose
  3.7-6 PostCompose
  3.7-7 PostCompose
  3.7-8 AddPostCompose
 3.8 Well-Definedness of Morphisms

  3.8-1 IsWellDefinedForMorphisms
  3.8-2 AddIsWellDefinedForMorphisms
 3.9 Basic Operations for Morphisms in Abelian Categories

  3.9-1 LiftAlongMonomorphism
  3.9-2 AddLiftAlongMonomorphism
  3.9-3 ColiftAlongEpimorphism
  3.9-4 AddColiftAlongEpimorphism
 3.10 Lift/ Colift

  3.10-1 Lift
  3.10-2 AddLift
  3.10-3 Colift
  3.10-4 AddColift
 3.11 Inverses

  3.11-1 AddInverse
 3.12 Tool functions for caches

  3.12-1 IsEqualForCacheForMorphisms
  3.12-2 AddIsEqualForCacheForMorphisms
4 Objects
 4.1 Attributes for the Type of Objects

  4.1-1 CapCategory
 4.2 Equality for Objects

  4.2-1 IsEqualForObjects
  4.2-2 AddIsEqualForObjects
 4.3 Categorical Properties of Objects

  4.3-1 AddIsProjective
  4.3-2 AddIsInjective
  4.3-3 AddIsTerminal
  4.3-4 AddIsInitial
  4.3-5 IsZeroForObjects
  4.3-6 AddIsZeroForObjects
 4.4 Tool functions for caches

  4.4-1 IsEqualForCacheForObjects
  4.4-2 AddIsEqualForCacheForObjects
 4.5 Well-Definedness of Objects

  4.5-1 IsWellDefinedForObjects
  4.5-2 AddIsWellDefinedForObjects
5 Category 2-Cells
 5.1 Attributes for the Type of 2-Cells

  5.1-1 Source
  5.1-2 Range
 5.2 Identity 2-Cell and Composition of 2-Cells

  5.2-1 IdentityTwoCell
  5.2-2 AddIdentityTwoCell
  5.2-3 HorizontalPreCompose
  5.2-4 AddHorizontalPreCompose
  5.2-5 HorizontalPostCompose
  5.2-6 AddHorizontalPostCompose
  5.2-7 VerticalPreCompose
  5.2-8 AddVerticalPreCompose
  5.2-9 VerticalPostCompose
  5.2-10 AddVerticalPostCompose
 5.3 Well-Definedness for 2-Cells

  5.3-1 IsWellDefinedForTwoCells
  5.3-2 AddIsWellDefinedForTwoCells
6 Universal Objects
 6.1 Kernel

  6.1-1 KernelObject
  6.1-2 KernelEmbedding
  6.1-3 KernelEmbeddingWithGivenKernelObject
  6.1-4 KernelLift
  6.1-5 KernelLiftWithGivenKernelObject
  6.1-6 AddKernelObject
  6.1-7 AddKernelEmbedding
  6.1-8 AddKernelEmbeddingWithGivenKernelObject
  6.1-9 AddKernelLift
  6.1-10 AddKernelLiftWithGivenKernelObject
  6.1-11 KernelObjectFunctorial
  6.1-12 KernelObjectFunctorial
  6.1-13 KernelObjectFunctorialWithGivenKernelObjects
  6.1-14 AddKernelObjectFunctorialWithGivenKernelObjects
 6.2 Cokernel

  6.2-1 CokernelObject
  6.2-2 CokernelProjection
  6.2-3 CokernelProjectionWithGivenCokernelObject
  6.2-4 CokernelColift
  6.2-5 CokernelColiftWithGivenCokernelObject
  6.2-6 AddCokernelObject
  6.2-7 AddCokernelProjection
  6.2-8 AddCokernelProjectionWithGivenCokernelObject
  6.2-9 AddCokernelColift
  6.2-10 AddCokernelColiftWithGivenCokernelObject
  6.2-11 CokernelObjectFunctorial
  6.2-12 CokernelObjectFunctorial
  6.2-13 CokernelObjectFunctorialWithGivenCokernelObjects
  6.2-14 AddCokernelObjectFunctorialWithGivenCokernelObjects
 6.3 Zero Object

  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 Terminal Object

  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 Initial Object

  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 Direct Sum

  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 MorphismBetweenDirectSumsOp
  6.6-24 AddProjectionInFactorOfDirectSum
  6.6-25 AddProjectionInFactorOfDirectSumWithGivenDirectSum
  6.6-26 AddInjectionOfCofactorOfDirectSum
  6.6-27 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum
  6.6-28 AddUniversalMorphismIntoDirectSum
  6.6-29 AddUniversalMorphismIntoDirectSumWithGivenDirectSum
  6.6-30 AddUniversalMorphismFromDirectSum
  6.6-31 AddUniversalMorphismFromDirectSumWithGivenDirectSum
  6.6-32 AddIsomorphismFromDirectSumToDirectProduct
  6.6-33 AddIsomorphismFromDirectProductToDirectSum
  6.6-34 AddIsomorphismFromDirectSumToCoproduct
  6.6-35 AddIsomorphismFromCoproductToDirectSum
  6.6-36 AddDirectSum
  6.6-37 DirectSumFunctorial
  6.6-38 DirectSumFunctorialWithGivenDirectSums
  6.6-39 AddDirectSumFunctorialWithGivenDirectSums
 6.7 Coproduct

  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 Direct Product

  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 Fiber Product

  6.9-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference
  6.9-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp
  6.9-3 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference
  6.9-4 IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
  6.9-5 IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp
  6.9-6 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
  6.9-7 DirectSumDiagonalDifference
  6.9-8 DirectSumDiagonalDifferenceOp
  6.9-9 AddDirectSumDiagonalDifference
  6.9-10 FiberProductEmbeddingInDirectSum
  6.9-11 FiberProductEmbeddingInDirectSumOp
  6.9-12 AddFiberProductEmbeddingInDirectSum
  6.9-13 FiberProduct
  6.9-14 FiberProductOp
  6.9-15 ProjectionInFactorOfFiberProduct
  6.9-16 ProjectionInFactorOfFiberProductOp
  6.9-17 ProjectionInFactorOfFiberProductWithGivenFiberProduct
  6.9-18 UniversalMorphismIntoFiberProduct
  6.9-19 UniversalMorphismIntoFiberProductOp
  6.9-20 UniversalMorphismIntoFiberProductWithGivenFiberProduct
  6.9-21 AddFiberProduct
  6.9-22 AddProjectionInFactorOfFiberProduct
  6.9-23 AddProjectionInFactorOfFiberProductWithGivenFiberProduct
  6.9-24 AddUniversalMorphismIntoFiberProduct
  6.9-25 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct
  6.9-26 FiberProductFunctorial
  6.9-27 FiberProductFunctorialWithGivenFiberProducts
  6.9-28 AddFiberProductFunctorialWithGivenFiberProducts
 6.10 Pushout

  6.10-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference
  6.10-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp
  6.10-3 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference
  6.10-4 IsomorphismFromCokernelOfDiagonalDifferenceToPushout
  6.10-5 IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp
  6.10-6 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout
  6.10-7 DirectSumCodiagonalDifference
  6.10-8 DirectSumCodiagonalDifferenceOp
  6.10-9 AddDirectSumCodiagonalDifference
  6.10-10 DirectSumProjectionInPushout
  6.10-11 DirectSumProjectionInPushoutOp
  6.10-12 AddDirectSumProjectionInPushout
  6.10-13 Pushout
  6.10-14 Pushout
  6.10-15 PushoutOp
  6.10-16 InjectionOfCofactorOfPushout
  6.10-17 InjectionOfCofactorOfPushoutOp
  6.10-18 InjectionOfCofactorOfPushoutWithGivenPushout
  6.10-19 UniversalMorphismFromPushout
  6.10-20 UniversalMorphismFromPushoutOp
  6.10-21 UniversalMorphismFromPushoutWithGivenPushout
  6.10-22 AddPushout
  6.10-23 AddInjectionOfCofactorOfPushout
  6.10-24 AddInjectionOfCofactorOfPushoutWithGivenPushout
  6.10-25 AddUniversalMorphismFromPushout
  6.10-26 AddUniversalMorphismFromPushoutWithGivenPushout
  6.10-27 PushoutFunctorial
  6.10-28 PushoutFunctorialWithGivenPushouts
  6.10-29 AddPushoutFunctorialWithGivenPushouts
 6.11 Image

  6.11-1 IsomorphismFromImageObjectToKernelOfCokernel
  6.11-2 AddIsomorphismFromImageObjectToKernelOfCokernel
  6.11-3 IsomorphismFromKernelOfCokernelToImageObject
  6.11-4 AddIsomorphismFromKernelOfCokernelToImageObject
  6.11-5 ImageObject
  6.11-6 ImageEmbedding
  6.11-7 ImageEmbeddingWithGivenImageObject
  6.11-8 CoastrictionToImage
  6.11-9 CoastrictionToImageWithGivenImageObject
  6.11-10 UniversalMorphismFromImage
  6.11-11 UniversalMorphismFromImageWithGivenImageObject
  6.11-12 AddImageObject
  6.11-13 AddImageEmbedding
  6.11-14 AddImageEmbeddingWithGivenImageObject
  6.11-15 AddCoastrictionToImage
  6.11-16 AddCoastrictionToImageWithGivenImageObject
  6.11-17 AddUniversalMorphismFromImage
  6.11-18 AddUniversalMorphismFromImageWithGivenImageObject
 6.12 Coimage

  6.12-1 MorphismFromCoimageToImage
  6.12-2 MorphismFromCoimageToImageWithGivenObjects
  6.12-3 AddMorphismFromCoimageToImageWithGivenObjects
  6.12-4 InverseMorphismFromCoimageToImage
  6.12-5 InverseMorphismFromCoimageToImageWithGivenObjects
  6.12-6 AddInverseMorphismFromCoimageToImageWithGivenObjects
  6.12-7 IsomorphismFromCoimageToCokernelOfKernel
  6.12-8 AddIsomorphismFromCoimageToCokernelOfKernel
  6.12-9 IsomorphismFromCokernelOfKernelToCoimage
  6.12-10 AddIsomorphismFromCokernelOfKernelToCoimage
  6.12-11 Coimage
  6.12-12 CoimageProjection
  6.12-13 CoimageProjection
  6.12-14 CoimageProjectionWithGivenCoimage
  6.12-15 AstrictionToCoimage
  6.12-16 AstrictionToCoimage
  6.12-17 AstrictionToCoimageWithGivenCoimage
  6.12-18 UniversalMorphismIntoCoimage
  6.12-19 UniversalMorphismIntoCoimageWithGivenCoimage
  6.12-20 AddCoimage
  6.12-21 AddCoimageProjection
  6.12-22 AddCoimageProjectionWithGivenCoimage
  6.12-23 AddAstrictionToCoimage
  6.12-24 AddAstrictionToCoimageWithGivenCoimage
  6.12-25 AddUniversalMorphismIntoCoimage
  6.12-26 AddUniversalMorphismIntoCoimageWithGivenCoimage
7 Tensor Product and Internal Hom
 7.1 Monoidal Categories

  7.1-1 TensorProductOnObjects
  7.1-2 AddTensorProductOnObjects
  7.1-3 TensorProductOnMorphisms
  7.1-4 TensorProductOnMorphismsWithGivenTensorProducts
  7.1-5 AddTensorProductOnMorphismsWithGivenTensorProducts
  7.1-6 AssociatorRightToLeft
  7.1-7 AssociatorRightToLeftWithGivenTensorProducts
  7.1-8 AddAssociatorRightToLeftWithGivenTensorProducts
  7.1-9 AssociatorLeftToRight
  7.1-10 AssociatorLeftToRightWithGivenTensorProducts
  7.1-11 AddAssociatorLeftToRightWithGivenTensorProducts
  7.1-12 TensorUnit
  7.1-13 AddTensorUnit
  7.1-14 LeftUnitor
  7.1-15 LeftUnitorWithGivenTensorProduct
  7.1-16 AddLeftUnitorWithGivenTensorProduct
  7.1-17 LeftUnitorInverse
  7.1-18 LeftUnitorInverseWithGivenTensorProduct
  7.1-19 AddLeftUnitorInverseWithGivenTensorProduct
  7.1-20 RightUnitor
  7.1-21 RightUnitorWithGivenTensorProduct
  7.1-22 AddRightUnitorWithGivenTensorProduct
  7.1-23 RightUnitorInverse
  7.1-24 RightUnitorInverseWithGivenTensorProduct
  7.1-25 AddRightUnitorInverseWithGivenTensorProduct
  7.1-26 LeftDistributivityExpanding
  7.1-27 LeftDistributivityExpandingWithGivenObjects
  7.1-28 AddLeftDistributivityExpandingWithGivenObjects
  7.1-29 LeftDistributivityFactoring
  7.1-30 LeftDistributivityFactoringWithGivenObjects
  7.1-31 AddLeftDistributivityFactoringWithGivenObjects
  7.1-32 RightDistributivityExpanding
  7.1-33 RightDistributivityExpandingWithGivenObjects
  7.1-34 AddRightDistributivityExpandingWithGivenObjects
  7.1-35 RightDistributivityFactoring
  7.1-36 RightDistributivityFactoringWithGivenObjects
  7.1-37 AddRightDistributivityFactoringWithGivenObjects
 7.2 Braided Monoidal Categories

  7.2-1 Braiding
  7.2-2 BraidingWithGivenTensorProducts
  7.2-3 AddBraidingWithGivenTensorProducts
  7.2-4 BraidingInverse
  7.2-5 BraidingInverseWithGivenTensorProducts
  7.2-6 AddBraidingInverseWithGivenTensorProducts
 7.3 Symmetric Monoidal Categories
 7.4 Symmetric Closed Monoidal Categories

  7.4-1 InternalHomOnObjects
  7.4-2 AddInternalHomOnObjects
  7.4-3 InternalHomOnMorphisms
  7.4-4 InternalHomOnMorphismsWithGivenInternalHoms
  7.4-5 AddInternalHomOnMorphismsWithGivenInternalHoms
  7.4-6 EvaluationMorphism
  7.4-7 EvaluationMorphismWithGivenSource
  7.4-8 AddEvaluationMorphismWithGivenSource
  7.4-9 CoevaluationMorphism
  7.4-10 CoevaluationMorphismWithGivenRange
  7.4-11 AddCoevaluationMorphismWithGivenRange
  7.4-12 TensorProductToInternalHomAdjunctionMap
  7.4-13 AddTensorProductToInternalHomAdjunctionMap
  7.4-14 InternalHomToTensorProductAdjunctionMap
  7.4-15 AddInternalHomToTensorProductAdjunctionMap
  7.4-16 MonoidalPreComposeMorphism
  7.4-17 MonoidalPreComposeMorphismWithGivenObjects
  7.4-18 AddMonoidalPreComposeMorphismWithGivenObjects
  7.4-19 MonoidalPostComposeMorphism
  7.4-20 MonoidalPostComposeMorphismWithGivenObjects
  7.4-21 AddMonoidalPostComposeMorphismWithGivenObjects
  7.4-22 DualOnObjects
  7.4-23 AddDualOnObjects
  7.4-24 DualOnMorphisms
  7.4-25 DualOnMorphismsWithGivenDuals
  7.4-26 AddDualOnMorphismsWithGivenDuals
  7.4-27 EvaluationForDual
  7.4-28 EvaluationForDualWithGivenTensorProduct
  7.4-29 AddEvaluationForDualWithGivenTensorProduct
  7.4-30 CoevaluationForDual
  7.4-31 CoevaluationForDualWithGivenTensorProduct
  7.4-32 AddCoevaluationForDualWithGivenTensorProduct
  7.4-33 MorphismToBidual
  7.4-34 MorphismToBidualWithGivenBidual
  7.4-35 AddMorphismToBidualWithGivenBidual
  7.4-36 TensorProductInternalHomCompatibilityMorphism
  7.4-37 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
  7.4-38 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
  7.4-39 TensorProductDualityCompatibilityMorphism
  7.4-40 TensorProductDualityCompatibilityMorphismWithGivenObjects
  7.4-41 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
  7.4-42 MorphismFromTensorProductToInternalHom
  7.4-43 MorphismFromTensorProductToInternalHomWithGivenObjects
  7.4-44 AddMorphismFromTensorProductToInternalHomWithGivenObjects
  7.4-45 IsomorphismFromTensorProductToInternalHom
  7.4-46 AddIsomorphismFromTensorProductToInternalHom
  7.4-47 MorphismFromInternalHomToTensorProduct
  7.4-48 MorphismFromInternalHomToTensorProductWithGivenObjects
  7.4-49 AddMorphismFromInternalHomToTensorProductWithGivenObjects
  7.4-50 IsomorphismFromInternalHomToTensorProduct
  7.4-51 AddIsomorphismFromInternalHomToTensorProduct
  7.4-52 TraceMap
  7.4-53 AddTraceMap
  7.4-54 RankMorphism
  7.4-55 AddRankMorphism
  7.4-56 IsomorphismFromDualToInternalHom
  7.4-57 AddIsomorphismFromDualToInternalHom
  7.4-58 IsomorphismFromInternalHomToDual
  7.4-59 AddIsomorphismFromInternalHomToDual
  7.4-60 UniversalPropertyOfDual
  7.4-61 AddUniversalPropertyOfDual
  7.4-62 LambdaIntroduction
  7.4-63 AddLambdaIntroduction
  7.4-64 LambdaElimination
  7.4-65 AddLambdaElimination
  7.4-66 IsomorphismFromObjectToInternalHom
  7.4-67 IsomorphismFromObjectToInternalHomWithGivenInternalHom
  7.4-68 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
  7.4-69 IsomorphismFromInternalHomToObject
  7.4-70 IsomorphismFromInternalHomToObjectWithGivenInternalHom
  7.4-71 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
 7.5 Rigid Symmetric Closed Monoidal Categories

  7.5-1 TensorProductInternalHomCompatibilityMorphismInverse
  7.5-2 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
  7.5-3 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
  7.5-4 MorphismFromBidual
  7.5-5 MorphismFromBidualWithGivenBidual
  7.5-6 AddMorphismFromBidualWithGivenBidual
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 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
 8.3 Derivation Graphs

  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 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 InstallDerivationsUsingOperation
  8.4-9 Reevaluate
  8.4-10 AddPrimitiveOperation
  8.4-11 PrintDerivationTree
  8.4-12 PrintTree
  8.4-13 PrintTreeRec
 8.5 Min Heaps for Strings

  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 Add Functions
 9.1 Functions Installed by Add
 9.2 Add Method
 9.3 InstallAdd Function
 9.4 Install All Adds
10 Technical Details
 10.1 The Category Cat

  10.1-1 ObjectCache
  10.1-2 MorphismCache
 10.2 Install Functions for IsWellDefined

  10.2-1 InstallMethodWithToDoForIsWellDefined
  10.2-2 InstallSetWithToDoForIsWellDefined
  10.2-3 DeclareAttributeWithToDoForIsWellDefined
  10.2-4 DeclareFamilyProperty
  10.2-5 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS
  10.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS
  10.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT
  10.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION
  10.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST
11 Examples and Tests
 11.1 Spectral Sequences
 11.2 Monoidal Categories
 11.3 Generalized Morphisms Category
 11.4 IsWellDefined
 11.5 Kernel
 11.6 FiberProduct
Index

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