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MonoidalCategories

Monoidal and monoidal (co)closed categories

2022.08-03

15 August 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 InternalHomToTensorProductAdjunctionMap

  1.5-10 MonoidalPreComposeMorphism

  1.5-11 MonoidalPreComposeMorphismWithGivenObjects

  1.5-12 MonoidalPostComposeMorphism

  1.5-13 MonoidalPostComposeMorphismWithGivenObjects

  1.5-14 DualOnObjects

  1.5-15 DualOnMorphisms

  1.5-16 DualOnMorphismsWithGivenDuals

  1.5-17 EvaluationForDual

  1.5-18 EvaluationForDualWithGivenTensorProduct

  1.5-19 MorphismToBidual

  1.5-20 MorphismToBidualWithGivenBidual

  1.5-21 TensorProductInternalHomCompatibilityMorphism

  1.5-22 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.5-23 TensorProductDualityCompatibilityMorphism

  1.5-24 TensorProductDualityCompatibilityMorphismWithGivenObjects

  1.5-25 MorphismFromTensorProductToInternalHom

  1.5-26 MorphismFromTensorProductToInternalHomWithGivenObjects

  1.5-27 IsomorphismFromDualObjectToInternalHomIntoTensorUnit

  1.5-28 IsomorphismFromInternalHomIntoTensorUnitToDualObject

  1.5-29 UniversalPropertyOfDual

  1.5-30 LambdaIntroduction

  1.5-31 LambdaElimination

  1.5-32 IsomorphismFromObjectToInternalHom

  1.5-33 IsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.5-34 IsomorphismFromInternalHomToObject

  1.5-35 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 InternalCoHomToTensorProductAdjunctionMap

  1.6-10 MonoidalPreCoComposeMorphism

  1.6-11 MonoidalPreCoComposeMorphismWithGivenObjects

  1.6-12 MonoidalPostCoComposeMorphism

  1.6-13 MonoidalPostCoComposeMorphismWithGivenObjects

  1.6-14 CoDualOnObjects

  1.6-15 CoDualOnMorphisms

  1.6-16 CoDualOnMorphismsWithGivenCoDuals

  1.6-17 CoclosedEvaluationForCoDual

  1.6-18 CoclosedEvaluationForCoDualWithGivenTensorProduct

  1.6-19 MorphismFromCoBidual

  1.6-20 MorphismFromCoBidualWithGivenCoBidual

  1.6-21 InternalCoHomTensorProductCompatibilityMorphism

  1.6-22 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

  1.6-23 CoDualityTensorProductCompatibilityMorphism

  1.6-24 CoDualityTensorProductCompatibilityMorphismWithGivenObjects

  1.6-25 MorphismFromInternalCoHomToTensorProduct

  1.6-26 MorphismFromInternalCoHomToTensorProductWithGivenObjects

  1.6-27 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

  1.6-28 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

  1.6-29 UniversalPropertyOfCoDual

  1.6-30 CoLambdaIntroduction

  1.6-31 CoLambdaElimination

  1.6-32 IsomorphismFromObjectToInternalCoHom

  1.6-33 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

  1.6-34 IsomorphismFromInternalCoHomToObject

  1.6-35 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 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit

  1.12-27 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject

  1.12-28 AddIsomorphismFromInternalHomToObject

  1.12-29 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom

  1.12-30 AddIsomorphismFromObjectToInternalHom

  1.12-31 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.12-32 AddLambdaElimination

  1.12-33 AddLambdaIntroduction

  1.12-34 AddMonoidalPostComposeMorphism

  1.12-35 AddMonoidalPostComposeMorphismWithGivenObjects

  1.12-36 AddMonoidalPreComposeMorphism

  1.12-37 AddMonoidalPreComposeMorphismWithGivenObjects

  1.12-38 AddMorphismFromTensorProductToInternalHom

  1.12-39 AddMorphismFromTensorProductToInternalHomWithGivenObjects

  1.12-40 AddMorphismToBidual

  1.12-41 AddMorphismToBidualWithGivenBidual

  1.12-42 AddTensorProductDualityCompatibilityMorphism

  1.12-43 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

  1.12-44 AddTensorProductInternalHomCompatibilityMorphism

  1.12-45 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.12-46 AddTensorProductToInternalHomAdjunctionMap

  1.12-47 AddUniversalPropertyOfDual

  1.12-48 AddCoDualOnMorphisms

  1.12-49 AddCoDualOnMorphismsWithGivenCoDuals

  1.12-50 AddCoDualOnObjects

  1.12-51 AddCoDualityTensorProductCompatibilityMorphism

  1.12-52 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects

  1.12-53 AddCoLambdaElimination

  1.12-54 AddCoLambdaIntroduction

  1.12-55 AddCoclosedCoevaluationMorphism

  1.12-56 AddCoclosedCoevaluationMorphismWithGivenSource

  1.12-57 AddCoclosedEvaluationForCoDual

  1.12-58 AddCoclosedEvaluationForCoDualWithGivenTensorProduct

  1.12-59 AddCoclosedEvaluationMorphism

  1.12-60 AddCoclosedEvaluationMorphismWithGivenRange

  1.12-61 AddInternalCoHomOnMorphisms

  1.12-62 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms

  1.12-63 AddInternalCoHomOnObjects

  1.12-64 AddInternalCoHomTensorProductCompatibilityMorphism

  1.12-65 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

  1.12-66 AddInternalCoHomToTensorProductAdjunctionMap

  1.12-67 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

  1.12-68 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

  1.12-69 AddIsomorphismFromInternalCoHomToObject

  1.12-70 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

  1.12-71 AddIsomorphismFromObjectToInternalCoHom

  1.12-72 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

  1.12-73 AddMonoidalPostCoComposeMorphism

  1.12-74 AddMonoidalPostCoComposeMorphismWithGivenObjects

  1.12-75 AddMonoidalPreCoComposeMorphism

  1.12-76 AddMonoidalPreCoComposeMorphismWithGivenObjects

  1.12-77 AddMorphismFromCoBidual

  1.12-78 AddMorphismFromCoBidualWithGivenCoBidual

  1.12-79 AddMorphismFromInternalCoHomToTensorProduct

  1.12-80 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects

  1.12-81 AddTensorProductToInternalCoHomAdjunctionMap

  1.12-82 AddUniversalPropertyOfCoDual

  1.12-83 AddAssociatorLeftToRight

  1.12-84 AddAssociatorLeftToRightWithGivenTensorProducts

  1.12-85 AddAssociatorRightToLeft

  1.12-86 AddAssociatorRightToLeftWithGivenTensorProducts

  1.12-87 AddLeftUnitor

  1.12-88 AddLeftUnitorInverse

  1.12-89 AddLeftUnitorInverseWithGivenTensorProduct

  1.12-90 AddLeftUnitorWithGivenTensorProduct

  1.12-91 AddRightUnitor

  1.12-92 AddRightUnitorInverse

  1.12-93 AddRightUnitorInverseWithGivenTensorProduct

  1.12-94 AddRightUnitorWithGivenTensorProduct

  1.12-95 AddTensorProductOnMorphisms

  1.12-96 AddTensorProductOnMorphismsWithGivenTensorProducts

  1.12-97 AddCoevaluationForDual

  1.12-98 AddCoevaluationForDualWithGivenTensorProduct

  1.12-99 AddIsomorphismFromInternalHomToTensorProductWithDualObject

  1.12-100 AddIsomorphismFromTensorProductWithDualObjectToInternalHom

  1.12-101 AddMorphismFromBidual

  1.12-102 AddMorphismFromBidualWithGivenBidual

  1.12-103 AddMorphismFromInternalHomToTensorProduct

  1.12-104 AddMorphismFromInternalHomToTensorProductWithGivenObjects

  1.12-105 AddRankMorphism

  1.12-106 AddTensorProductInternalHomCompatibilityMorphismInverse

  1.12-107 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

  1.12-108 AddTraceMap

  1.12-109 AddCoRankMorphism

  1.12-110 AddCoTraceMap

  1.12-111 AddCoclosedCoevaluationForCoDual

  1.12-112 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct

  1.12-113 AddInternalCoHomTensorProductCompatibilityMorphismInverse

  1.12-114 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

  1.12-115 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject

  1.12-116 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

  1.12-117 AddMorphismFromTensorProductToInternalCoHom

  1.12-118 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects

  1.12-119 AddMorphismToCoBidual

  1.12-120 AddMorphismToCoBidualWithGivenCoBidual
2 Examples and Tests
3 Code Generation for Monodial Categories
4 The terminal category with multiple objects
Index

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