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MonoidalCategories

Monoidal and monoidal (co)closed categories

2026.06-01

2 June 2026

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: http://algebra.mathematik.uni-siegen.de/barakat/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Sebastian Gutsche
Email: sebastian.gutsche@gmail.com
Homepage: http://algebra.mathematik.uni-siegen.de/gutsche/
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

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

Tom Kuhmichel
Email: tom.kuhmichel@student.uni-siegen.de
Homepage: https://github.com/TKuh
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

Fabian Zickgraf
Email: f.zickgraf@dashdos.com
Homepage: https://github.com/zickgraf/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Monoidal Categories
 1.1 Monoidal Categories

  1.1-1 TensorProductOnMorphisms
  1.1-2 TensorProductOnMorphismsWithGivenTensorProducts
  1.1-3 TensorProductOnMorphismAndObject
  1.1-4 TensorProductOnMorphismAndObjectWithGivenTensorProducts
  1.1-5 TensorProductOnObjectAndMorphism
  1.1-6 TensorProductOnObjectAndMorphismWithGivenTensorProducts
  1.1-7 AssociatorRightToLeft
  1.1-8 AssociatorRightToLeftWithGivenTensorProducts
  1.1-9 AssociatorLeftToRight
  1.1-10 AssociatorLeftToRightWithGivenTensorProducts
  1.1-11 LeftUnitor
  1.1-12 LeftUnitorWithGivenTensorProduct
  1.1-13 LeftUnitorInverse
  1.1-14 LeftUnitorInverseWithGivenTensorProduct
  1.1-15 RightUnitor
  1.1-16 RightUnitorWithGivenTensorProduct
  1.1-17 RightUnitorInverse
  1.1-18 RightUnitorInverseWithGivenTensorProduct
  1.1-19 TensorProductOnObjects
  1.1-20 TensorUnit
 1.2 Additive Monoidal Categories

  1.2-1 LeftDistributivityExpanding
  1.2-2 LeftDistributivityExpandingWithGivenObjects
  1.2-3 LeftDistributivityExpandingUsingMultiplicities
  1.2-4 LeftDistributivityExpandingUsingMultiplicitiesWithGivenObjects
  1.2-5 LeftDistributivityFactoring
  1.2-6 LeftDistributivityFactoringWithGivenObjects
  1.2-7 LeftDistributivityFactoringUsingMultiplicities
  1.2-8 LeftDistributivityFactoringUsingMultiplicitiesWithGivenObjects
  1.2-9 RightDistributivityExpanding
  1.2-10 RightDistributivityExpandingWithGivenObjects
  1.2-11 RightDistributivityExpandingUsingMultiplicities
  1.2-12 RightDistributivityExpandingUsingMultiplicitiesWithGivenObjects
  1.2-13 RightDistributivityFactoring
  1.2-14 RightDistributivityFactoringWithGivenObjects
  1.2-15 RightDistributivityFactoringUsingMultiplicities
  1.2-16 RightDistributivityFactoringUsingMultiplicitiesWithGivenObjects
 1.3 Braided Monoidal Categories

  1.3-1 Braiding
  1.3-2 BraidingWithGivenTensorProducts
  1.3-3 BraidingInverse
  1.3-4 BraidingInverseWithGivenTensorProducts
 1.4 Symmetric Monoidal Categories
 1.5 Left Closed Monoidal Categories

  1.5-1 LeftInternalHomOnObjects
  1.5-2 LeftInternalHomOnMorphisms
  1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms
  1.5-4 LeftClosedMonoidalEvaluationMorphism
  1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource
  1.5-6 LeftClosedMonoidalCoevaluationMorphism
  1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange
  1.5-8 TensorProductToLeftInternalHomAdjunctMorphism
  1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom
  1.5-10 LeftInternalHomToTensorProductAdjunctMorphism
  1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct
  1.5-12 LeftClosedMonoidalPreComposeMorphism
  1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects
  1.5-14 LeftClosedMonoidalPostComposeMorphism
  1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects
  1.5-16 LeftDualOnObjects
  1.5-17 LeftDualOnMorphisms
  1.5-18 LeftDualOnMorphismsWithGivenLeftDuals
  1.5-19 LeftClosedMonoidalEvaluationForLeftDual
  1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct
  1.5-21 MorphismToLeftBidual
  1.5-22 MorphismToLeftBidualWithGivenLeftBidual
  1.5-23 TensorProductLeftInternalHomCompatibilityMorphism
  1.5-24 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects
  1.5-25 TensorProductLeftDualityCompatibilityMorphism
  1.5-26 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects
  1.5-27 MorphismFromTensorProductToLeftInternalHom
  1.5-28 MorphismFromTensorProductToLeftInternalHomWithGivenObjects
  1.5-29 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit
  1.5-30 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject
  1.5-31 UniversalPropertyOfLeftDual
  1.5-32 LeftClosedMonoidalLambdaIntroduction
  1.5-33 LeftClosedMonoidalLambdaElimination
  1.5-34 IsomorphismFromObjectToLeftInternalHom
  1.5-35 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom
  1.5-36 IsomorphismFromLeftInternalHomToObject
  1.5-37 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom
 1.6 Closed Monoidal Categories

  1.6-1 InternalHomOnObjects
  1.6-2 InternalHomOnMorphisms
  1.6-3 InternalHomOnMorphismsWithGivenInternalHoms
  1.6-4 ClosedMonoidalRightEvaluationMorphism
  1.6-5 ClosedMonoidalRightEvaluationMorphismWithGivenSource
  1.6-6 ClosedMonoidalRightCoevaluationMorphism
  1.6-7 ClosedMonoidalRightCoevaluationMorphismWithGivenRange
  1.6-8 TensorProductToInternalHomRightAdjunctMorphism
  1.6-9 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom
  1.6-10 TensorProductToInternalHomRightAdjunctionIsomorphism
  1.6-11 TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects
  1.6-12 InternalHomToTensorProductRightAdjunctMorphism
  1.6-13 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
  1.6-14 InternalHomToTensorProductRightAdjunctionIsomorphism
  1.6-15 InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects
  1.6-16 ClosedMonoidalLeftEvaluationMorphism
  1.6-17 ClosedMonoidalLeftEvaluationMorphismWithGivenSource
  1.6-18 ClosedMonoidalLeftCoevaluationMorphism
  1.6-19 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange
  1.6-20 TensorProductToInternalHomLeftAdjunctMorphism
  1.6-21 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom
  1.6-22 TensorProductToInternalHomLeftAdjunctionIsomorphism
  1.6-23 TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects
  1.6-24 InternalHomToTensorProductLeftAdjunctMorphism
  1.6-25 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
  1.6-26 InternalHomToTensorProductLeftAdjunctionIsomorphism
  1.6-27 InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects
  1.6-28 MonoidalPreComposeMorphism
  1.6-29 MonoidalPreComposeMorphismWithGivenObjects
  1.6-30 MonoidalPostComposeMorphism
  1.6-31 MonoidalPostComposeMorphismWithGivenObjects
  1.6-32 DualOnObjects
  1.6-33 DualOnMorphisms
  1.6-34 DualOnMorphismsWithGivenDuals
  1.6-35 EvaluationForDual
  1.6-36 EvaluationForDualWithGivenTensorProduct
  1.6-37 MorphismToBidual
  1.6-38 MorphismToBidualWithGivenBidual
  1.6-39 TensorProductInternalHomCompatibilityMorphism
  1.6-40 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
  1.6-41 TensorProductDualityCompatibilityMorphism
  1.6-42 TensorProductDualityCompatibilityMorphismWithGivenObjects
  1.6-43 MorphismFromTensorProductToInternalHom
  1.6-44 MorphismFromTensorProductToInternalHomWithGivenObjects
  1.6-45 IsomorphismFromDualObjectToInternalHomIntoTensorUnit
  1.6-46 IsomorphismFromInternalHomIntoTensorUnitToDualObject
  1.6-47 UniversalPropertyOfDual
  1.6-48 LambdaIntroduction
  1.6-49 LambdaElimination
  1.6-50 IsomorphismFromObjectToInternalHom
  1.6-51 IsomorphismFromObjectToInternalHomWithGivenInternalHom
  1.6-52 IsomorphismFromInternalHomToObject
  1.6-53 IsomorphismFromInternalHomToObjectWithGivenInternalHom
 1.7 Left Coclosed Monoidal Categories

  1.7-1 LeftInternalCoHomOnObjects
  1.7-2 LeftInternalCoHomOnMorphisms
  1.7-3 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms
  1.7-4 LeftCoclosedMonoidalEvaluationMorphism
  1.7-5 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange
  1.7-6 LeftCoclosedMonoidalCoevaluationMorphism
  1.7-7 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource
  1.7-8 TensorProductToLeftInternalCoHomAdjunctMorphism
  1.7-9 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom
  1.7-10 LeftInternalCoHomToTensorProductAdjunctMorphism
  1.7-11 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct
  1.7-12 LeftCoclosedMonoidalPreCoComposeMorphism
  1.7-13 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects
  1.7-14 LeftCoclosedMonoidalPostCoComposeMorphism
  1.7-15 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects
  1.7-16 LeftCoDualOnObjects
  1.7-17 LeftCoDualOnMorphisms
  1.7-18 LeftCoDualOnMorphismsWithGivenLeftCoDuals
  1.7-19 LeftCoclosedMonoidalEvaluationForLeftCoDual
  1.7-20 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct
  1.7-21 MorphismFromLeftCoBidual
  1.7-22 MorphismFromLeftCoBidualWithGivenLeftCoBidual
  1.7-23 LeftInternalCoHomTensorProductCompatibilityMorphism
  1.7-24 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
  1.7-25 LeftCoDualityTensorProductCompatibilityMorphism
  1.7-26 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects
  1.7-27 MorphismFromLeftInternalCoHomToTensorProduct
  1.7-28 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects
  1.7-29 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit
  1.7-30 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject
  1.7-31 UniversalPropertyOfLeftCoDual
  1.7-32 LeftCoclosedMonoidalLambdaIntroduction
  1.7-33 LeftCoclosedMonoidalLambdaElimination
  1.7-34 IsomorphismFromObjectToLeftInternalCoHom
  1.7-35 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom
  1.7-36 IsomorphismFromLeftInternalCoHomToObject
  1.7-37 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom
 1.8 Coclosed Monoidal Categories

  1.8-1 InternalCoHomOnObjects
  1.8-2 InternalCoHomOnMorphisms
  1.8-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms
  1.8-4 CoclosedMonoidalRightEvaluationMorphism
  1.8-5 CoclosedMonoidalRightEvaluationMorphismWithGivenRange
  1.8-6 CoclosedMonoidalRightCoevaluationMorphism
  1.8-7 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource
  1.8-8 TensorProductToInternalCoHomRightAdjunctMorphism
  1.8-9 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom
  1.8-10 InternalCoHomToTensorProductRightAdjunctMorphism
  1.8-11 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
  1.8-12 CoclosedMonoidalLeftEvaluationMorphism
  1.8-13 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange
  1.8-14 CoclosedMonoidalLeftCoevaluationMorphism
  1.8-15 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource
  1.8-16 TensorProductToInternalCoHomLeftAdjunctMorphism
  1.8-17 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom
  1.8-18 InternalCoHomToTensorProductLeftAdjunctMorphism
  1.8-19 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
  1.8-20 MonoidalPreCoComposeMorphism
  1.8-21 MonoidalPreCoComposeMorphismWithGivenObjects
  1.8-22 MonoidalPostCoComposeMorphism
  1.8-23 MonoidalPostCoComposeMorphismWithGivenObjects
  1.8-24 CoDualOnObjects
  1.8-25 CoDualOnMorphisms
  1.8-26 CoDualOnMorphismsWithGivenCoDuals
  1.8-27 CoclosedEvaluationForCoDual
  1.8-28 CoclosedEvaluationForCoDualWithGivenTensorProduct
  1.8-29 MorphismFromCoBidual
  1.8-30 MorphismFromCoBidualWithGivenCoBidual
  1.8-31 InternalCoHomTensorProductCompatibilityMorphism
  1.8-32 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
  1.8-33 CoDualityTensorProductCompatibilityMorphism
  1.8-34 CoDualityTensorProductCompatibilityMorphismWithGivenObjects
  1.8-35 MorphismFromInternalCoHomToTensorProduct
  1.8-36 MorphismFromInternalCoHomToTensorProductWithGivenObjects
  1.8-37 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
  1.8-38 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
  1.8-39 UniversalPropertyOfCoDual
  1.8-40 CoLambdaIntroduction
  1.8-41 CoLambdaElimination
  1.8-42 IsomorphismFromObjectToInternalCoHom
  1.8-43 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
  1.8-44 IsomorphismFromInternalCoHomToObject
  1.8-45 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
 1.9 Symmetric Closed Monoidal Categories
 1.10 Symmetric Coclosed Monoidal Categories
 1.11 Rigid Symmetric Closed Monoidal Categories

  1.11-1 IsomorphismFromTensorProductWithDualObjectToInternalHom
  1.11-2 IsomorphismFromInternalHomToTensorProductWithDualObject
  1.11-3 MorphismFromInternalHomToTensorProduct
  1.11-4 MorphismFromInternalHomToTensorProductWithGivenObjects
  1.11-5 TensorProductInternalHomCompatibilityMorphismInverse
  1.11-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
  1.11-7 CoevaluationForDual
  1.11-8 CoevaluationForDualWithGivenTensorProduct
  1.11-9 TraceMap
  1.11-10 RankMorphism
  1.11-11 MorphismFromBidual
  1.11-12 MorphismFromBidualWithGivenBidual
 1.12 Rigid Symmetric Coclosed Monoidal Categories

  1.12-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
  1.12-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
  1.12-3 MorphismFromTensorProductToInternalCoHom
  1.12-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects
  1.12-5 InternalCoHomTensorProductCompatibilityMorphismInverse
  1.12-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
  1.12-7 CoclosedCoevaluationForCoDual
  1.12-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct
  1.12-9 CoTraceMap
  1.12-10 CoRankMorphism
  1.12-11 MorphismToCoBidual
  1.12-12 MorphismToCoBidualWithGivenCoBidual
 1.13 Convenience Methods

  1.13-1 InternalHom
  1.13-2 InternalCoHom
  1.13-3 LeftInternalHom
  1.13-4 LeftInternalCoHom
 1.14 Add-methods

  1.14-1 AddLeftDistributivityExpanding
  1.14-2 AddLeftDistributivityExpandingUsingMultiplicities
  1.14-3 AddLeftDistributivityExpandingUsingMultiplicitiesWithGivenObjects
  1.14-4 AddLeftDistributivityExpandingWithGivenObjects
  1.14-5 AddLeftDistributivityFactoring
  1.14-6 AddLeftDistributivityFactoringUsingMultiplicities
  1.14-7 AddLeftDistributivityFactoringUsingMultiplicitiesWithGivenObjects
  1.14-8 AddLeftDistributivityFactoringWithGivenObjects
  1.14-9 AddRightDistributivityExpanding
  1.14-10 AddRightDistributivityExpandingUsingMultiplicities
  1.14-11 AddRightDistributivityExpandingUsingMultiplicitiesWithGivenObjects
  1.14-12 AddRightDistributivityExpandingWithGivenObjects
  1.14-13 AddRightDistributivityFactoring
  1.14-14 AddRightDistributivityFactoringUsingMultiplicities
  1.14-15 AddRightDistributivityFactoringUsingMultiplicitiesWithGivenObjects
  1.14-16 AddRightDistributivityFactoringWithGivenObjects
  1.14-17 AddBraiding
  1.14-18 AddBraidingInverse
  1.14-19 AddBraidingInverseWithGivenTensorProducts
  1.14-20 AddBraidingWithGivenTensorProducts
  1.14-21 AddClosedMonoidalLeftCoevaluationMorphism
  1.14-22 AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange
  1.14-23 AddClosedMonoidalLeftEvaluationMorphism
  1.14-24 AddClosedMonoidalLeftEvaluationMorphismWithGivenSource
  1.14-25 AddClosedMonoidalRightCoevaluationMorphism
  1.14-26 AddClosedMonoidalRightCoevaluationMorphismWithGivenRange
  1.14-27 AddClosedMonoidalRightEvaluationMorphism
  1.14-28 AddClosedMonoidalRightEvaluationMorphismWithGivenSource
  1.14-29 AddDualOnMorphisms
  1.14-30 AddDualOnMorphismsWithGivenDuals
  1.14-31 AddDualOnObjects
  1.14-32 AddEvaluationForDual
  1.14-33 AddEvaluationForDualWithGivenTensorProduct
  1.14-34 AddInternalHomOnMorphisms
  1.14-35 AddInternalHomOnMorphismsWithGivenInternalHoms
  1.14-36 AddInternalHomOnObjects
  1.14-37 AddInternalHomToTensorProductLeftAdjunctMorphism
  1.14-38 AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
  1.14-39 AddInternalHomToTensorProductLeftAdjunctionIsomorphism
  1.14-40 AddInternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects
  1.14-41 AddInternalHomToTensorProductRightAdjunctMorphism
  1.14-42 AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
  1.14-43 AddInternalHomToTensorProductRightAdjunctionIsomorphism
  1.14-44 AddInternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects
  1.14-45 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit
  1.14-46 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject
  1.14-47 AddIsomorphismFromInternalHomToObject
  1.14-48 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
  1.14-49 AddIsomorphismFromObjectToInternalHom
  1.14-50 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
  1.14-51 AddLambdaElimination
  1.14-52 AddLambdaIntroduction
  1.14-53 AddMonoidalPostComposeMorphism
  1.14-54 AddMonoidalPostComposeMorphismWithGivenObjects
  1.14-55 AddMonoidalPreComposeMorphism
  1.14-56 AddMonoidalPreComposeMorphismWithGivenObjects
  1.14-57 AddMorphismFromTensorProductToInternalHom
  1.14-58 AddMorphismFromTensorProductToInternalHomWithGivenObjects
  1.14-59 AddMorphismToBidual
  1.14-60 AddMorphismToBidualWithGivenBidual
  1.14-61 AddTensorProductDualityCompatibilityMorphism
  1.14-62 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
  1.14-63 AddTensorProductInternalHomCompatibilityMorphism
  1.14-64 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
  1.14-65 AddTensorProductToInternalHomLeftAdjunctMorphism
  1.14-66 AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom
  1.14-67 AddTensorProductToInternalHomLeftAdjunctionIsomorphism
  1.14-68 AddTensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects
  1.14-69 AddTensorProductToInternalHomRightAdjunctMorphism
  1.14-70 AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom
  1.14-71 AddTensorProductToInternalHomRightAdjunctionIsomorphism
  1.14-72 AddTensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects
  1.14-73 AddUniversalPropertyOfDual
  1.14-74 AddCoDualOnMorphisms
  1.14-75 AddCoDualOnMorphismsWithGivenCoDuals
  1.14-76 AddCoDualOnObjects
  1.14-77 AddCoDualityTensorProductCompatibilityMorphism
  1.14-78 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects
  1.14-79 AddCoLambdaElimination
  1.14-80 AddCoLambdaIntroduction
  1.14-81 AddCoclosedEvaluationForCoDual
  1.14-82 AddCoclosedEvaluationForCoDualWithGivenTensorProduct
  1.14-83 AddCoclosedMonoidalLeftCoevaluationMorphism
  1.14-84 AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource
  1.14-85 AddCoclosedMonoidalLeftEvaluationMorphism
  1.14-86 AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange
  1.14-87 AddCoclosedMonoidalRightCoevaluationMorphism
  1.14-88 AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource
  1.14-89 AddCoclosedMonoidalRightEvaluationMorphism
  1.14-90 AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange
  1.14-91 AddInternalCoHomOnMorphisms
  1.14-92 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms
  1.14-93 AddInternalCoHomOnObjects
  1.14-94 AddInternalCoHomTensorProductCompatibilityMorphism
  1.14-95 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
  1.14-96 AddInternalCoHomToTensorProductLeftAdjunctMorphism
  1.14-97 AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
  1.14-98 AddInternalCoHomToTensorProductRightAdjunctMorphism
  1.14-99 AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
  1.14-100 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
  1.14-101 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
  1.14-102 AddIsomorphismFromInternalCoHomToObject
  1.14-103 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
  1.14-104 AddIsomorphismFromObjectToInternalCoHom
  1.14-105 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
  1.14-106 AddMonoidalPostCoComposeMorphism
  1.14-107 AddMonoidalPostCoComposeMorphismWithGivenObjects
  1.14-108 AddMonoidalPreCoComposeMorphism
  1.14-109 AddMonoidalPreCoComposeMorphismWithGivenObjects
  1.14-110 AddMorphismFromCoBidual
  1.14-111 AddMorphismFromCoBidualWithGivenCoBidual
  1.14-112 AddMorphismFromInternalCoHomToTensorProduct
  1.14-113 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects
  1.14-114 AddTensorProductToInternalCoHomLeftAdjunctMorphism
  1.14-115 AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom
  1.14-116 AddTensorProductToInternalCoHomRightAdjunctMorphism
  1.14-117 AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom
  1.14-118 AddUniversalPropertyOfCoDual
  1.14-119 AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit
  1.14-120 AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject
  1.14-121 AddIsomorphismFromLeftInternalHomToObject
  1.14-122 AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom
  1.14-123 AddIsomorphismFromObjectToLeftInternalHom
  1.14-124 AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom
  1.14-125 AddLeftClosedMonoidalCoevaluationMorphism
  1.14-126 AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange
  1.14-127 AddLeftClosedMonoidalEvaluationForLeftDual
  1.14-128 AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct
  1.14-129 AddLeftClosedMonoidalEvaluationMorphism
  1.14-130 AddLeftClosedMonoidalEvaluationMorphismWithGivenSource
  1.14-131 AddLeftClosedMonoidalLambdaElimination
  1.14-132 AddLeftClosedMonoidalLambdaIntroduction
  1.14-133 AddLeftClosedMonoidalPostComposeMorphism
  1.14-134 AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects
  1.14-135 AddLeftClosedMonoidalPreComposeMorphism
  1.14-136 AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects
  1.14-137 AddLeftDualOnMorphisms
  1.14-138 AddLeftDualOnMorphismsWithGivenLeftDuals
  1.14-139 AddLeftDualOnObjects
  1.14-140 AddLeftInternalHomOnMorphisms
  1.14-141 AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms
  1.14-142 AddLeftInternalHomOnObjects
  1.14-143 AddLeftInternalHomToTensorProductAdjunctMorphism
  1.14-144 AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct
  1.14-145 AddMorphismFromTensorProductToLeftInternalHom
  1.14-146 AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects
  1.14-147 AddMorphismToLeftBidual
  1.14-148 AddMorphismToLeftBidualWithGivenLeftBidual
  1.14-149 AddTensorProductLeftDualityCompatibilityMorphism
  1.14-150 AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects
  1.14-151 AddTensorProductLeftInternalHomCompatibilityMorphism
  1.14-152 AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects
  1.14-153 AddTensorProductToLeftInternalHomAdjunctMorphism
  1.14-154 AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom
  1.14-155 AddUniversalPropertyOfLeftDual
  1.14-156 AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit
  1.14-157 AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject
  1.14-158 AddIsomorphismFromLeftInternalCoHomToObject
  1.14-159 AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom
  1.14-160 AddIsomorphismFromObjectToLeftInternalCoHom
  1.14-161 AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom
  1.14-162 AddLeftCoDualOnMorphisms
  1.14-163 AddLeftCoDualOnMorphismsWithGivenLeftCoDuals
  1.14-164 AddLeftCoDualOnObjects
  1.14-165 AddLeftCoDualityTensorProductCompatibilityMorphism
  1.14-166 AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects
  1.14-167 AddLeftCoclosedMonoidalCoevaluationMorphism
  1.14-168 AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource
  1.14-169 AddLeftCoclosedMonoidalEvaluationForLeftCoDual
  1.14-170 AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct
  1.14-171 AddLeftCoclosedMonoidalEvaluationMorphism
  1.14-172 AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange
  1.14-173 AddLeftCoclosedMonoidalLambdaElimination
  1.14-174 AddLeftCoclosedMonoidalLambdaIntroduction
  1.14-175 AddLeftCoclosedMonoidalPostCoComposeMorphism
  1.14-176 AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects
  1.14-177 AddLeftCoclosedMonoidalPreCoComposeMorphism
  1.14-178 AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects
  1.14-179 AddLeftInternalCoHomOnMorphisms
  1.14-180 AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms
  1.14-181 AddLeftInternalCoHomOnObjects
  1.14-182 AddLeftInternalCoHomTensorProductCompatibilityMorphism
  1.14-183 AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
  1.14-184 AddLeftInternalCoHomToTensorProductAdjunctMorphism
  1.14-185 AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct
  1.14-186 AddMorphismFromLeftCoBidual
  1.14-187 AddMorphismFromLeftCoBidualWithGivenLeftCoBidual
  1.14-188 AddMorphismFromLeftInternalCoHomToTensorProduct
  1.14-189 AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects
  1.14-190 AddTensorProductToLeftInternalCoHomAdjunctMorphism
  1.14-191 AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom
  1.14-192 AddUniversalPropertyOfLeftCoDual
  1.14-193 AddAssociatorLeftToRight
  1.14-194 AddAssociatorLeftToRightWithGivenTensorProducts
  1.14-195 AddAssociatorRightToLeft
  1.14-196 AddAssociatorRightToLeftWithGivenTensorProducts
  1.14-197 AddLeftUnitor
  1.14-198 AddLeftUnitorInverse
  1.14-199 AddLeftUnitorInverseWithGivenTensorProduct
  1.14-200 AddLeftUnitorWithGivenTensorProduct
  1.14-201 AddRightUnitor
  1.14-202 AddRightUnitorInverse
  1.14-203 AddRightUnitorInverseWithGivenTensorProduct
  1.14-204 AddRightUnitorWithGivenTensorProduct
  1.14-205 AddTensorProductOnMorphismAndObject
  1.14-206 AddTensorProductOnMorphismAndObjectWithGivenTensorProducts
  1.14-207 AddTensorProductOnMorphisms
  1.14-208 AddTensorProductOnMorphismsWithGivenTensorProducts
  1.14-209 AddTensorProductOnObjectAndMorphism
  1.14-210 AddTensorProductOnObjectAndMorphismWithGivenTensorProducts
  1.14-211 AddTensorProductOnObjects
  1.14-212 AddTensorUnit
  1.14-213 AddCoevaluationForDual
  1.14-214 AddCoevaluationForDualWithGivenTensorProduct
  1.14-215 AddIsomorphismFromInternalHomToTensorProductWithDualObject
  1.14-216 AddIsomorphismFromTensorProductWithDualObjectToInternalHom
  1.14-217 AddMorphismFromBidual
  1.14-218 AddMorphismFromBidualWithGivenBidual
  1.14-219 AddMorphismFromInternalHomToTensorProduct
  1.14-220 AddMorphismFromInternalHomToTensorProductWithGivenObjects
  1.14-221 AddRankMorphism
  1.14-222 AddTensorProductInternalHomCompatibilityMorphismInverse
  1.14-223 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
  1.14-224 AddTraceMap
  1.14-225 AddCoRankMorphism
  1.14-226 AddCoTraceMap
  1.14-227 AddCoclosedCoevaluationForCoDual
  1.14-228 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct
  1.14-229 AddInternalCoHomTensorProductCompatibilityMorphismInverse
  1.14-230 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
  1.14-231 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject
  1.14-232 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
  1.14-233 AddMorphismFromTensorProductToInternalCoHom
  1.14-234 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects
  1.14-235 AddMorphismToCoBidual
  1.14-236 AddMorphismToCoBidualWithGivenCoBidual
2 Examples and Tests
 2.1 Test functions

  2.1-1 AdditiveMonoidalCategoriesTest
  2.1-2 TestBraidingForInvertibility
  2.1-3 TestBraidingCompatibility
  2.1-4 TestBraidingCompatibilityForAllTriplesInList
  2.1-5 BraidedMonoidalCategoriesTest
  2.1-6 ClosedMonoidalCategoriesTest
  2.1-7 ClosedMonoidalCategoriesTestWithGiven
  2.1-8 CoclosedMonoidalCategoriesTest
  2.1-9 CoclosedMonoidalCategoriesTestWithGiven
  2.1-10 LeftClosedMonoidalCategoriesTest
  2.1-11 LeftClosedMonoidalCategoriesTestWithGiven
  2.1-12 LeftCoclosedMonoidalCategoriesTest
  2.1-13 LeftCoclosedMonoidalCategoriesTestWithGiven
  2.1-14 MonoidalCategoriesTensorProductOnObjectsAndTensorUnitTest
  2.1-15 TestMonoidalUnitorsForInvertibility
  2.1-16 TestAssociatorForInvertibility
  2.1-17 TestMonoidalTriangleIdentity
  2.1-18 TestMonoidalTriangleIdentityForAllPairsInList
  2.1-19 TestMonoidalPentagonIdentity
  2.1-20 TestMonoidalPentagonIdentityUsingWithGivenOperations
  2.1-21 TestMonoidalPentagonIdentityForAllQuadruplesInList
  2.1-22 MonoidalCategoriesTest
  2.1-23 TestZigZagIdentitiesForDual
  2.1-24 RigidSymmetricClosedMonoidalCategoriesTest
  2.1-25 RigidSymmetricCoclosedMonoidalCategoriesTest
3 Code Generation for Monodial Categories
 3.1 Monoidal Categories

  3.1-1 WriteFileForMonoidalStructure
 3.2 Closed Monoidal Categories

  3.2-1 WriteFileForClosedMonoidalStructure
  3.2-2 WriteFileForLeftClosedMonoidalStructure
 3.3 Coclosed Monoidal Categories

  3.3-1 WriteFileForCoclosedMonoidalStructure
  3.3-2 WriteFileForLeftCoclosedMonoidalStructure
4 The terminal category with multiple objects
 4.1 Constructors
 4.2 GAP Categories
5 Legacy Operations and Synonyms
 5.1 Legacy operations

  5.1-1 CoclosedCoevaluationMorphism
  5.1-2 CoclosedCoevaluationMorphismWithGivenSource
  5.1-3 CoclosedEvaluationMorphism
  5.1-4 CoclosedEvaluationMorphismWithGivenRange
  5.1-5 CoevaluationMorphism
  5.1-6 CoevaluationMorphismWithGivenRange
 5.2 Synonyms for legacy operations

  5.2-1 EvaluationMorphism
  5.2-2 EvaluationMorphismWithGivenSource
  5.2-3 InternalCoHomToTensorProductAdjunctionMap
  5.2-4 InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct
  5.2-5 InternalHomToTensorProductAdjunctionMap
  5.2-6 InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct
  5.2-7 TensorProductToInternalCoHomAdjunctionMap
  5.2-8 TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom
  5.2-9 TensorProductToInternalHomAdjunctionMap
  5.2-10 TensorProductToInternalHomAdjunctionMapWithGivenInternalHom
  5.2-11 InternalCoHomToTensorProductLeftAdjunctionMap
  5.2-12 InternalHomToTensorProductLeftAdjunctionMap
  5.2-13 TensorProductToInternalCoHomLeftAdjunctionMap
  5.2-14 TensorProductToInternalCoHomLeftAdjunctionMapWithGivenInternalCoHom
  5.2-15 TensorProductToInternalHomLeftAdjunctionMap
  5.2-16 TensorProductToInternalHomLeftAdjunctionMapWithGivenInternalHom
6 MonoidalCategories automatic generated documentation
 6.1 MonoidalCategories automatic generated documentation of properties

  6.1-1 IsAdditiveMonoidalCategory
  6.1-2 IsBraidedMonoidalCategory
  6.1-3 IsClosedMonoidalCategory
  6.1-4 IsCoclosedMonoidalCategory
  6.1-5 IsLeftClosedMonoidalCategory
  6.1-6 IsLeftCoclosedMonoidalCategory
  6.1-7 IsMonoidalCategory
  6.1-8 IsStrictMonoidalCategory
  6.1-9 IsRigidSymmetricClosedMonoidalCategory
  6.1-10 IsRigidSymmetricCoclosedMonoidalCategory
  6.1-11 IsSymmetricClosedMonoidalCategory
  6.1-12 IsSymmetricCoclosedMonoidalCategory
  6.1-13 IsSymmetricMonoidalCategory
Index

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