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CAP

Categories, Algorithms, Programming

2016.02.19

19/02/2016

Sebastian Gutsche
Email: gutsche@mathematik.uni-kl.de
Homepage: http://wwwb.math.rwth-aachen.de/~gutsche
Address:
TODO

Sebastian Posur
Email: sposur@momo.math.rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/Mitarbeiter/posur.php
Address:
Lehrstuhl B für Mathematik RWTH - Aachen
Templergraben 64
52062 Aachen
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 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 AddDirectSumFunctorial

  6.6-39 DirectSumFunctorialOp
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 Examples and Tests
Index

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