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MonoidalCategories

Monoidal and monoidal (co)closed categories

2019.06.07

7 June 2019

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

Contents

1 Tensor Product and Internal Hom
 1.1 Monoidal Categories

  1.1-1 TensorProductOnObjects

  1.1-2 AddTensorProductOnObjects

  1.1-3 TensorProductOnMorphisms

  1.1-4 TensorProductOnMorphismsWithGivenTensorProducts

  1.1-5 AddTensorProductOnMorphismsWithGivenTensorProducts

  1.1-6 AssociatorRightToLeft

  1.1-7 AssociatorRightToLeftWithGivenTensorProducts

  1.1-8 AddAssociatorRightToLeftWithGivenTensorProducts

  1.1-9 AssociatorLeftToRight

  1.1-10 AssociatorLeftToRightWithGivenTensorProducts

  1.1-11 AddAssociatorLeftToRightWithGivenTensorProducts

  1.1-12 TensorUnit

  1.1-13 AddTensorUnit

  1.1-14 LeftUnitor

  1.1-15 LeftUnitorWithGivenTensorProduct

  1.1-16 AddLeftUnitorWithGivenTensorProduct

  1.1-17 LeftUnitorInverse

  1.1-18 LeftUnitorInverseWithGivenTensorProduct

  1.1-19 AddLeftUnitorInverseWithGivenTensorProduct

  1.1-20 RightUnitor

  1.1-21 RightUnitorWithGivenTensorProduct

  1.1-22 AddRightUnitorWithGivenTensorProduct

  1.1-23 RightUnitorInverse

  1.1-24 RightUnitorInverseWithGivenTensorProduct

  1.1-25 AddRightUnitorInverseWithGivenTensorProduct

  1.1-26 LeftDistributivityExpanding

  1.1-27 LeftDistributivityExpandingWithGivenObjects

  1.1-28 AddLeftDistributivityExpandingWithGivenObjects

  1.1-29 LeftDistributivityFactoring

  1.1-30 LeftDistributivityFactoringWithGivenObjects

  1.1-31 AddLeftDistributivityFactoringWithGivenObjects

  1.1-32 RightDistributivityExpanding

  1.1-33 RightDistributivityExpandingWithGivenObjects

  1.1-34 AddRightDistributivityExpandingWithGivenObjects

  1.1-35 RightDistributivityFactoring

  1.1-36 RightDistributivityFactoringWithGivenObjects

  1.1-37 AddRightDistributivityFactoringWithGivenObjects
 1.4 Symmetric Closed Monoidal Categories

  1.4-1 InternalHomOnObjects

  1.4-2 AddInternalHomOnObjects

  1.4-3 InternalHomOnMorphisms

  1.4-4 InternalHomOnMorphismsWithGivenInternalHoms

  1.4-5 AddInternalHomOnMorphismsWithGivenInternalHoms

  1.4-6 EvaluationMorphism

  1.4-7 EvaluationMorphismWithGivenSource

  1.4-8 AddEvaluationMorphismWithGivenSource

  1.4-9 CoevaluationMorphism

  1.4-10 CoevaluationMorphismWithGivenRange

  1.4-11 AddCoevaluationMorphismWithGivenRange

  1.4-12 TensorProductToInternalHomAdjunctionMap

  1.4-13 AddTensorProductToInternalHomAdjunctionMap

  1.4-14 InternalHomToTensorProductAdjunctionMap

  1.4-15 AddInternalHomToTensorProductAdjunctionMap

  1.4-16 MonoidalPreComposeMorphism

  1.4-17 MonoidalPreComposeMorphismWithGivenObjects

  1.4-18 AddMonoidalPreComposeMorphismWithGivenObjects

  1.4-19 MonoidalPostComposeMorphism

  1.4-20 MonoidalPostComposeMorphismWithGivenObjects

  1.4-21 AddMonoidalPostComposeMorphismWithGivenObjects

  1.4-22 DualOnObjects

  1.4-23 AddDualOnObjects

  1.4-24 DualOnMorphisms

  1.4-25 DualOnMorphismsWithGivenDuals

  1.4-26 AddDualOnMorphismsWithGivenDuals

  1.4-27 EvaluationForDual

  1.4-28 EvaluationForDualWithGivenTensorProduct

  1.4-29 AddEvaluationForDualWithGivenTensorProduct

  1.4-30 CoevaluationForDual

  1.4-31 CoevaluationForDualWithGivenTensorProduct

  1.4-32 AddCoevaluationForDualWithGivenTensorProduct

  1.4-33 MorphismToBidual

  1.4-34 MorphismToBidualWithGivenBidual

  1.4-35 AddMorphismToBidualWithGivenBidual

  1.4-36 TensorProductInternalHomCompatibilityMorphism

  1.4-37 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.4-38 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

  1.4-39 TensorProductDualityCompatibilityMorphism

  1.4-40 TensorProductDualityCompatibilityMorphismWithGivenObjects

  1.4-41 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

  1.4-42 MorphismFromTensorProductToInternalHom

  1.4-43 MorphismFromTensorProductToInternalHomWithGivenObjects

  1.4-44 AddMorphismFromTensorProductToInternalHomWithGivenObjects

  1.4-45 IsomorphismFromTensorProductToInternalHom

  1.4-46 AddIsomorphismFromTensorProductToInternalHom

  1.4-47 MorphismFromInternalHomToTensorProduct

  1.4-48 MorphismFromInternalHomToTensorProductWithGivenObjects

  1.4-49 AddMorphismFromInternalHomToTensorProductWithGivenObjects

  1.4-50 IsomorphismFromInternalHomToTensorProduct

  1.4-51 AddIsomorphismFromInternalHomToTensorProduct

  1.4-52 TraceMap

  1.4-53 AddTraceMap

  1.4-54 RankMorphism

  1.4-55 AddRankMorphism

  1.4-56 IsomorphismFromDualToInternalHom

  1.4-57 AddIsomorphismFromDualToInternalHom

  1.4-58 IsomorphismFromInternalHomToDual

  1.4-59 AddIsomorphismFromInternalHomToDual

  1.4-60 UniversalPropertyOfDual

  1.4-61 AddUniversalPropertyOfDual

  1.4-62 LambdaIntroduction

  1.4-63 AddLambdaIntroduction

  1.4-64 LambdaElimination

  1.4-65 AddLambdaElimination

  1.4-66 IsomorphismFromObjectToInternalHom

  1.4-67 IsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.4-68 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

  1.4-69 IsomorphismFromInternalHomToObject

  1.4-70 IsomorphismFromInternalHomToObjectWithGivenInternalHom

  1.4-71 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
2 Examples and Tests
Index

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