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CAP

Categories, Algorithms, Programming

2018.08.31

31 August 2018

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: https://sebastianpos.github.io
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
2 Category of Categories
3 Morphisms
4 Objects
5 Category 2-Cells
6 Universal Objects
 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 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
 6.11 Fiber Product

  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 UniversalMorphismIntoFiberProduct

  6.11-25 UniversalMorphismIntoFiberProductOp

  6.11-26 UniversalMorphismIntoFiberProductWithGivenFiberProduct

  6.11-27 AddFiberProduct

  6.11-28 AddProjectionInFactorOfFiberProduct

  6.11-29 AddProjectionInFactorOfFiberProductWithGivenFiberProduct

  6.11-30 AddUniversalMorphismIntoFiberProduct

  6.11-31 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct

  6.11-32 FiberProductFunctorial

  6.11-33 FiberProductFunctorialWithGivenFiberProducts

  6.11-34 AddFiberProductFunctorialWithGivenFiberProducts
 6.12 Pushout

  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 UniversalMorphismFromPushout

  6.12-26 UniversalMorphismFromPushoutOp

  6.12-27 UniversalMorphismFromPushoutWithGivenPushout

  6.12-28 AddPushout

  6.12-29 AddInjectionOfCofactorOfPushout

  6.12-30 AddInjectionOfCofactorOfPushoutWithGivenPushout

  6.12-31 AddUniversalMorphismFromPushout

  6.12-32 AddUniversalMorphismFromPushoutWithGivenPushout

  6.12-33 PushoutFunctorial

  6.12-34 PushoutFunctorialWithGivenPushouts

  6.12-35 AddPushoutFunctorialWithGivenPushouts
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
 7.4 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
8 Managing Derived Methods
9 Add Functions
10 Technical Details
11 Monoidal Categories
12 Prepare functions
13 Examples and Tests
Index

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