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MonoidalCategories

Monoidal and monoidal (co)closed categories

2022.11-05

25 November 2022

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: gutsche@mathematik.uni-siegen.de
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-siegen.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

Contents

1 Monoidal Categories
 1.5 Closed Monoidal Categories

  1.5-1 InternalHomOnObjects

  1.5-2 InternalHomOnMorphisms

  1.5-3 InternalHomOnMorphismsWithGivenInternalHoms

  1.5-4 EvaluationMorphism

  1.5-5 EvaluationMorphismWithGivenSource

  1.5-6 CoevaluationMorphism

  1.5-7 CoevaluationMorphismWithGivenRange

  1.5-8 TensorProductToInternalHomAdjunctionMap

  1.5-9 TensorProductToInternalHomAdjunctionMapWithGivenInternalHom

  1.5-10 InternalHomToTensorProductAdjunctionMap

  1.5-11 InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct

  1.5-12 MonoidalPreComposeMorphism

  1.5-13 MonoidalPreComposeMorphismWithGivenObjects

  1.5-14 MonoidalPostComposeMorphism

  1.5-15 MonoidalPostComposeMorphismWithGivenObjects

  1.5-16 DualOnObjects

  1.5-17 DualOnMorphisms

  1.5-18 DualOnMorphismsWithGivenDuals

  1.5-19 EvaluationForDual

  1.5-20 EvaluationForDualWithGivenTensorProduct

  1.5-21 MorphismToBidual

  1.5-22 MorphismToBidualWithGivenBidual

  1.5-23 TensorProductInternalHomCompatibilityMorphism

  1.5-24 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.5-25 TensorProductDualityCompatibilityMorphism

  1.5-26 TensorProductDualityCompatibilityMorphismWithGivenObjects

  1.5-27 MorphismFromTensorProductToInternalHom

  1.5-28 MorphismFromTensorProductToInternalHomWithGivenObjects

  1.5-29 IsomorphismFromDualObjectToInternalHomIntoTensorUnit

  1.5-30 IsomorphismFromInternalHomIntoTensorUnitToDualObject

  1.5-31 UniversalPropertyOfDual

  1.5-32 LambdaIntroduction

  1.5-33 LambdaElimination

  1.5-34 IsomorphismFromObjectToInternalHom

  1.5-35 IsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.5-36 IsomorphismFromInternalHomToObject

  1.5-37 IsomorphismFromInternalHomToObjectWithGivenInternalHom
 1.6 Coclosed Monoidal Categories

  1.6-1 InternalCoHomOnObjects

  1.6-2 InternalCoHomOnMorphisms

  1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms

  1.6-4 CoclosedEvaluationMorphism

  1.6-5 CoclosedEvaluationMorphismWithGivenRange

  1.6-6 CoclosedCoevaluationMorphism

  1.6-7 CoclosedCoevaluationMorphismWithGivenSource

  1.6-8 TensorProductToInternalCoHomAdjunctionMap

  1.6-9 TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom

  1.6-10 InternalCoHomToTensorProductAdjunctionMap

  1.6-11 InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct

  1.6-12 MonoidalPreCoComposeMorphism

  1.6-13 MonoidalPreCoComposeMorphismWithGivenObjects

  1.6-14 MonoidalPostCoComposeMorphism

  1.6-15 MonoidalPostCoComposeMorphismWithGivenObjects

  1.6-16 CoDualOnObjects

  1.6-17 CoDualOnMorphisms

  1.6-18 CoDualOnMorphismsWithGivenCoDuals

  1.6-19 CoclosedEvaluationForCoDual

  1.6-20 CoclosedEvaluationForCoDualWithGivenTensorProduct

  1.6-21 MorphismFromCoBidual

  1.6-22 MorphismFromCoBidualWithGivenCoBidual

  1.6-23 InternalCoHomTensorProductCompatibilityMorphism

  1.6-24 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

  1.6-25 CoDualityTensorProductCompatibilityMorphism

  1.6-26 CoDualityTensorProductCompatibilityMorphismWithGivenObjects

  1.6-27 MorphismFromInternalCoHomToTensorProduct

  1.6-28 MorphismFromInternalCoHomToTensorProductWithGivenObjects

  1.6-29 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

  1.6-30 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

  1.6-31 UniversalPropertyOfCoDual

  1.6-32 CoLambdaIntroduction

  1.6-33 CoLambdaElimination

  1.6-34 IsomorphismFromObjectToInternalCoHom

  1.6-35 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

  1.6-36 IsomorphismFromInternalCoHomToObject

  1.6-37 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
 1.12 Add-methods

  1.12-1 AddLeftDistributivityExpanding

  1.12-2 AddLeftDistributivityExpandingWithGivenObjects

  1.12-3 AddLeftDistributivityFactoring

  1.12-4 AddLeftDistributivityFactoringWithGivenObjects

  1.12-5 AddRightDistributivityExpanding

  1.12-6 AddRightDistributivityExpandingWithGivenObjects

  1.12-7 AddRightDistributivityFactoring

  1.12-8 AddRightDistributivityFactoringWithGivenObjects

  1.12-9 AddBraiding

  1.12-10 AddBraidingInverse

  1.12-11 AddBraidingInverseWithGivenTensorProducts

  1.12-12 AddBraidingWithGivenTensorProducts

  1.12-13 AddCoevaluationMorphism

  1.12-14 AddCoevaluationMorphismWithGivenRange

  1.12-15 AddDualOnMorphisms

  1.12-16 AddDualOnMorphismsWithGivenDuals

  1.12-17 AddDualOnObjects

  1.12-18 AddEvaluationForDual

  1.12-19 AddEvaluationForDualWithGivenTensorProduct

  1.12-20 AddEvaluationMorphism

  1.12-21 AddEvaluationMorphismWithGivenSource

  1.12-22 AddInternalHomOnMorphisms

  1.12-23 AddInternalHomOnMorphismsWithGivenInternalHoms

  1.12-24 AddInternalHomOnObjects

  1.12-25 AddInternalHomToTensorProductAdjunctionMap

  1.12-26 AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct

  1.12-27 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit

  1.12-28 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject

  1.12-29 AddIsomorphismFromInternalHomToObject

  1.12-30 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom

  1.12-31 AddIsomorphismFromObjectToInternalHom

  1.12-32 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.12-33 AddLambdaElimination

  1.12-34 AddLambdaIntroduction

  1.12-35 AddMonoidalPostComposeMorphism

  1.12-36 AddMonoidalPostComposeMorphismWithGivenObjects

  1.12-37 AddMonoidalPreComposeMorphism

  1.12-38 AddMonoidalPreComposeMorphismWithGivenObjects

  1.12-39 AddMorphismFromTensorProductToInternalHom

  1.12-40 AddMorphismFromTensorProductToInternalHomWithGivenObjects

  1.12-41 AddMorphismToBidual

  1.12-42 AddMorphismToBidualWithGivenBidual

  1.12-43 AddTensorProductDualityCompatibilityMorphism

  1.12-44 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

  1.12-45 AddTensorProductInternalHomCompatibilityMorphism

  1.12-46 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.12-47 AddTensorProductToInternalHomAdjunctionMap

  1.12-48 AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom

  1.12-49 AddUniversalPropertyOfDual

  1.12-50 AddCoDualOnMorphisms

  1.12-51 AddCoDualOnMorphismsWithGivenCoDuals

  1.12-52 AddCoDualOnObjects

  1.12-53 AddCoDualityTensorProductCompatibilityMorphism

  1.12-54 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects

  1.12-55 AddCoLambdaElimination

  1.12-56 AddCoLambdaIntroduction

  1.12-57 AddCoclosedCoevaluationMorphism

  1.12-58 AddCoclosedCoevaluationMorphismWithGivenSource

  1.12-59 AddCoclosedEvaluationForCoDual

  1.12-60 AddCoclosedEvaluationForCoDualWithGivenTensorProduct

  1.12-61 AddCoclosedEvaluationMorphism

  1.12-62 AddCoclosedEvaluationMorphismWithGivenRange

  1.12-63 AddInternalCoHomOnMorphisms

  1.12-64 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms

  1.12-65 AddInternalCoHomOnObjects

  1.12-66 AddInternalCoHomTensorProductCompatibilityMorphism

  1.12-67 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

  1.12-68 AddInternalCoHomToTensorProductAdjunctionMap

  1.12-69 AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct

  1.12-70 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

  1.12-71 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

  1.12-72 AddIsomorphismFromInternalCoHomToObject

  1.12-73 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

  1.12-74 AddIsomorphismFromObjectToInternalCoHom

  1.12-75 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

  1.12-76 AddMonoidalPostCoComposeMorphism

  1.12-77 AddMonoidalPostCoComposeMorphismWithGivenObjects

  1.12-78 AddMonoidalPreCoComposeMorphism

  1.12-79 AddMonoidalPreCoComposeMorphismWithGivenObjects

  1.12-80 AddMorphismFromCoBidual

  1.12-81 AddMorphismFromCoBidualWithGivenCoBidual

  1.12-82 AddMorphismFromInternalCoHomToTensorProduct

  1.12-83 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects

  1.12-84 AddTensorProductToInternalCoHomAdjunctionMap

  1.12-85 AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom

  1.12-86 AddUniversalPropertyOfCoDual

  1.12-87 AddAssociatorLeftToRight

  1.12-88 AddAssociatorLeftToRightWithGivenTensorProducts

  1.12-89 AddAssociatorRightToLeft

  1.12-90 AddAssociatorRightToLeftWithGivenTensorProducts

  1.12-91 AddLeftUnitor

  1.12-92 AddLeftUnitorInverse

  1.12-93 AddLeftUnitorInverseWithGivenTensorProduct

  1.12-94 AddLeftUnitorWithGivenTensorProduct

  1.12-95 AddRightUnitor

  1.12-96 AddRightUnitorInverse

  1.12-97 AddRightUnitorInverseWithGivenTensorProduct

  1.12-98 AddRightUnitorWithGivenTensorProduct

  1.12-99 AddTensorProductOnMorphisms

  1.12-100 AddTensorProductOnMorphismsWithGivenTensorProducts

  1.12-101 AddCoevaluationForDual

  1.12-102 AddCoevaluationForDualWithGivenTensorProduct

  1.12-103 AddIsomorphismFromInternalHomToTensorProductWithDualObject

  1.12-104 AddIsomorphismFromTensorProductWithDualObjectToInternalHom

  1.12-105 AddMorphismFromBidual

  1.12-106 AddMorphismFromBidualWithGivenBidual

  1.12-107 AddMorphismFromInternalHomToTensorProduct

  1.12-108 AddMorphismFromInternalHomToTensorProductWithGivenObjects

  1.12-109 AddRankMorphism

  1.12-110 AddTensorProductInternalHomCompatibilityMorphismInverse

  1.12-111 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

  1.12-112 AddTraceMap

  1.12-113 AddCoRankMorphism

  1.12-114 AddCoTraceMap

  1.12-115 AddCoclosedCoevaluationForCoDual

  1.12-116 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct

  1.12-117 AddInternalCoHomTensorProductCompatibilityMorphismInverse

  1.12-118 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

  1.12-119 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject

  1.12-120 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

  1.12-121 AddMorphismFromTensorProductToInternalCoHom

  1.12-122 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects

  1.12-123 AddMorphismToCoBidual

  1.12-124 AddMorphismToCoBidualWithGivenCoBidual
2 Examples and Tests
3 Code Generation for Monodial Categories
4 The terminal category with multiple objects
5 MonoidalCategories automatic generated documentation
Index

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