  
  
                              [1X MonoidalCategories [101X
  
  
                 [1X Monoidal and monoidal (co)closed categories [101X
  
  
                                   2022.12-01
  
  
                                14 December 2022
  
  
                                Mohamed Barakat
  
                               Sebastian Gutsche
  
                                Sebastian Posur
  
                                 Tom Kuhmichel
  
  
  
  Mohamed Barakat
      Email:    [7Xmailto:mohamed.barakat@uni-siegen.de[107X
      Homepage: [7Xhttp://algebra.mathematik.uni-siegen.de/barakat/[107X
      Address:  [33X[0;14YWalter-Flex-Str. 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:gutsche@mathematik.uni-siegen.de[107X
      Homepage: [7Xhttp://algebra.mathematik.uni-siegen.de/gutsche/[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Sebastian Posur
      Email:    [7Xmailto:sebastian.posur@uni-siegen.de[107X
      Homepage: [7Xhttp://algebra.mathematik.uni-siegen.de/posur/[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Tom Kuhmichel
      Email:    [7Xmailto:tom.kuhmichel@student.uni-siegen.de[107X
      Homepage: [7Xhttps://github.com/TKuh[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57068 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  
  -------------------------------------------------------
  
  
  [1XContents (MonoidalCategories)[101X
  
  1 [33X[0;0YMonoidal Categories[133X
    1.1 [33X[0;0YMonoidal Categories[133X
      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 [33X[0;0YAdditive Monoidal Categories[133X
      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 [33X[0;0YBraided Monoidal Categories[133X
      1.3-1 Braiding
      1.3-2 BraidingWithGivenTensorProducts
      1.3-3 BraidingInverse
      1.3-4 BraidingInverseWithGivenTensorProducts
    1.4 [33X[0;0YSymmetric Monoidal Categories[133X
    1.5 [33X[0;0YClosed Monoidal Categories[133X
      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 [33X[0;0YCoclosed Monoidal Categories[133X
      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 [33X[0;0YSymmetric Closed Monoidal Categories[133X
    1.8 [33X[0;0YSymmetric Coclosed Monoidal Categories[133X
    1.9 [33X[0;0YRigid Symmetric Closed Monoidal Categories[133X
      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 [33X[0;0YRigid Symmetric Coclosed Monoidal Categories[133X
      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 [33X[0;0YConvenience Methods[133X
      1.11-1 InternalHom
      1.11-2 InternalCoHom
    1.12 [33X[0;0YAdd-methods[133X
      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 [33X[0;0YExamples and Tests[133X
    2.1 [33X[0;0YTest functions[133X
      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 [33X[0;0YBasics[133X
  3 [33X[0;0YCode Generation for Monodial Categories[133X
    3.1 [33X[0;0YMonoidal Categories[133X
      3.1-1 WriteFileForMonoidalStructure
    3.2 [33X[0;0YClosed Monoidal Categories[133X
      3.2-1 WriteFileForClosedMonoidalStructure
    3.3 [33X[0;0YCoclosed Monoidal Categories[133X
      3.3-1 WriteFileForCoclosedMonoidalStructure
  4 [33X[0;0YThe terminal category with multiple objects[133X
    4.1 [33X[0;0YConstructors[133X
  5 [33X[0;0YMonoidalCategories automatic generated documentation[133X
    5.1 [33X[0;0YMonoidalCategories automatic generated documentation of properties[133X
      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
  
  
  [32X
