Goto Chapter: Top 1 2 3 4 5 Ind
 [Top of Book]  [Contents]   [Next Chapter] 

MonoidalCategories

Monoidal and monoidal (co)closed categories

2022.12-01

14 December 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
 1.1 Monoidal Categories

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

  1.2-1 LeftDistributivityExpanding
  1.2-2 LeftDistributivityExpandingWithGivenObjects
  1.2-3 LeftDistributivityFactoring
  1.2-4 LeftDistributivityFactoringWithGivenObjects
  1.2-5 RightDistributivityExpanding
  1.2-6 RightDistributivityExpandingWithGivenObjects
  1.2-7 RightDistributivityFactoring
  1.2-8 RightDistributivityFactoringWithGivenObjects
 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 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
 1.7 Symmetric Closed Monoidal Categories
 1.8 Symmetric Coclosed Monoidal Categories
 1.9 Rigid Symmetric Closed Monoidal Categories

  1.9-1 IsomorphismFromTensorProductWithDualObjectToInternalHom
  1.9-2 IsomorphismFromInternalHomToTensorProductWithDualObject
  1.9-3 MorphismFromInternalHomToTensorProduct
  1.9-4 MorphismFromInternalHomToTensorProductWithGivenObjects
  1.9-5 TensorProductInternalHomCompatibilityMorphismInverse
  1.9-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
  1.9-7 CoevaluationForDual
  1.9-8 CoevaluationForDualWithGivenTensorProduct
  1.9-9 TraceMap
  1.9-10 RankMorphism
  1.9-11 MorphismFromBidual
  1.9-12 MorphismFromBidualWithGivenBidual
 1.10 Rigid Symmetric Coclosed Monoidal Categories

  1.10-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
  1.10-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
  1.10-3 MorphismFromTensorProductToInternalCoHom
  1.10-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects
  1.10-5 InternalCoHomTensorProductCompatibilityMorphismInverse
  1.10-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
  1.10-7 CoclosedCoevaluationForCoDual
  1.10-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct
  1.10-9 CoTraceMap
  1.10-10 CoRankMorphism
  1.10-11 MorphismToCoBidual
  1.10-12 MorphismToCoBidualWithGivenCoBidual
 1.11 Convenience Methods

  1.11-1 InternalHom
  1.11-2 InternalCoHom
 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
 2.1 Test functions

  2.1-1 AdditiveMonoidalCategoriesTest
  2.1-2 BraidedMonoidalCategoriesTest
  2.1-3 ClosedMonoidalCategoriesTest
  2.1-4 CoclosedMonoidalCategoriesTest
  2.1-5 MonoidalCategoriesTensorProductAndUnitTest
  2.1-6 MonoidalCategoriesTest
  2.1-7 RigidSymmetricClosedMonoidalCategoriesTest
  2.1-8 RigidSymmetricCoclosedMonoidalCategoriesTest
 2.2 Basics
3 Code Generation for Monodial Categories
 3.1 Monoidal Categories

  3.1-1 WriteFileForMonoidalStructure
 3.2 Closed Monoidal Categories

  3.2-1 WriteFileForClosedMonoidalStructure
 3.3 Coclosed Monoidal Categories

  3.3-1 WriteFileForCoclosedMonoidalStructure
4 The terminal category with multiple objects
 4.1 Constructors
5 MonoidalCategories automatic generated documentation
 5.1 MonoidalCategories automatic generated documentation of properties

  5.1-1 IsBraidedMonoidalCategory
  5.1-2 IsClosedMonoidalCategory
  5.1-3 IsCoclosedMonoidalCategory
  5.1-4 IsMonoidalCategory
  5.1-5 IsStrictMonoidalCategory
  5.1-6 IsRigidSymmetricClosedMonoidalCategory
  5.1-7 IsRigidSymmetricCoclosedMonoidalCategory
  5.1-8 IsSymmetricClosedMonoidalCategory
  5.1-9 IsSymmetricCoclosedMonoidalCategory
  5.1-10 IsSymmetricMonoidalCategory
Index

 [Top of Book]  [Contents]   [Next Chapter] 
Goto Chapter: Top 1 2 3 4 5 Ind

generated by GAPDoc2HTML