  
  
                              [1X MonoidalCategories [101X
  
  
                 [1X Monoidal and monoidal (co)closed categories [101X
  
  
                                   2022.08-03
  
  
                                 15 August 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 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 [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 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
    1.7 [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 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 [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
      4.1-1 TerminalCategory
      4.1-2 TerminalCategoryWithMultipleObjects
      4.1-3 /
    4.2 [33X[0;0Y[5XGAP[105X Categories[133X
      4.2-1 IsTerminalCategoryWithMultipleObjects
      4.2-2 IsCellInTerminalCategoryWithMultipleObjects
      4.2-3 IsObjectInTerminalCategoryWithMultipleObjects
      4.2-4 IsMorphismInTerminalCategoryWithMultipleObjects
  
  
  [32X
