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7 Tensor Product and Internal Hom
 7.1 Monoidal Categories

  7.1-1 TensorProductOnObjects

  7.1-2 AddTensorProductOnObjects

  7.1-3 TensorProductOnMorphisms

  7.1-4 TensorProductOnMorphismsWithGivenTensorProducts

  7.1-5 AddTensorProductOnMorphismsWithGivenTensorProducts

  7.1-6 AssociatorRightToLeft

  7.1-7 AssociatorRightToLeftWithGivenTensorProducts

  7.1-8 AddAssociatorRightToLeftWithGivenTensorProducts

  7.1-9 AssociatorLeftToRight

  7.1-10 AssociatorLeftToRightWithGivenTensorProducts

  7.1-11 AddAssociatorLeftToRightWithGivenTensorProducts

  7.1-12 TensorUnit

  7.1-13 AddTensorUnit

  7.1-14 LeftUnitor

  7.1-15 LeftUnitorWithGivenTensorProduct

  7.1-16 AddLeftUnitorWithGivenTensorProduct

  7.1-17 LeftUnitorInverse

  7.1-18 LeftUnitorInverseWithGivenTensorProduct

  7.1-19 AddLeftUnitorInverseWithGivenTensorProduct

  7.1-20 RightUnitor

  7.1-21 RightUnitorWithGivenTensorProduct

  7.1-22 AddRightUnitorWithGivenTensorProduct

  7.1-23 RightUnitorInverse

  7.1-24 RightUnitorInverseWithGivenTensorProduct

  7.1-25 AddRightUnitorInverseWithGivenTensorProduct

  7.1-26 LeftDistributivityExpanding

  7.1-27 LeftDistributivityExpandingWithGivenObjects

  7.1-28 AddLeftDistributivityExpandingWithGivenObjects

  7.1-29 LeftDistributivityFactoring

  7.1-30 LeftDistributivityFactoringWithGivenObjects

  7.1-31 AddLeftDistributivityFactoringWithGivenObjects

  7.1-32 RightDistributivityExpanding

  7.1-33 RightDistributivityExpandingWithGivenObjects

  7.1-34 AddRightDistributivityExpandingWithGivenObjects

  7.1-35 RightDistributivityFactoring

  7.1-36 RightDistributivityFactoringWithGivenObjects

  7.1-37 AddRightDistributivityFactoringWithGivenObjects
 7.2 Braided Monoidal Categories
 7.3 Symmetric Monoidal Categories
 7.4 Symmetric Closed Monoidal Categories

  7.4-1 InternalHomOnObjects

  7.4-2 AddInternalHomOnObjects

  7.4-3 InternalHomOnMorphisms

  7.4-4 InternalHomOnMorphismsWithGivenInternalHoms

  7.4-5 AddInternalHomOnMorphismsWithGivenInternalHoms

  7.4-6 EvaluationMorphism

  7.4-7 EvaluationMorphismWithGivenSource

  7.4-8 AddEvaluationMorphismWithGivenSource

  7.4-9 CoevaluationMorphism

  7.4-10 CoevaluationMorphismWithGivenRange

  7.4-11 AddCoevaluationMorphismWithGivenRange

  7.4-12 TensorProductToInternalHomAdjunctionMap

  7.4-13 AddTensorProductToInternalHomAdjunctionMap

  7.4-14 InternalHomToTensorProductAdjunctionMap

  7.4-15 AddInternalHomToTensorProductAdjunctionMap

  7.4-16 MonoidalPreComposeMorphism

  7.4-17 MonoidalPreComposeMorphismWithGivenObjects

  7.4-18 AddMonoidalPreComposeMorphismWithGivenObjects

  7.4-19 MonoidalPostComposeMorphism

  7.4-20 MonoidalPostComposeMorphismWithGivenObjects

  7.4-21 AddMonoidalPostComposeMorphismWithGivenObjects

  7.4-22 DualOnObjects

  7.4-23 AddDualOnObjects

  7.4-24 DualOnMorphisms

  7.4-25 DualOnMorphismsWithGivenDuals

  7.4-26 AddDualOnMorphismsWithGivenDuals

  7.4-27 EvaluationForDual

  7.4-28 EvaluationForDualWithGivenTensorProduct

  7.4-29 AddEvaluationForDualWithGivenTensorProduct

  7.4-30 CoevaluationForDual

  7.4-31 CoevaluationForDualWithGivenTensorProduct

  7.4-32 AddCoevaluationForDualWithGivenTensorProduct

  7.4-33 MorphismToBidual

  7.4-34 MorphismToBidualWithGivenBidual

  7.4-35 AddMorphismToBidualWithGivenBidual

  7.4-36 TensorProductInternalHomCompatibilityMorphism

  7.4-37 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

  7.4-38 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

  7.4-39 TensorProductDualityCompatibilityMorphism

  7.4-40 TensorProductDualityCompatibilityMorphismWithGivenObjects

  7.4-41 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

  7.4-42 MorphismFromTensorProductToInternalHom

  7.4-43 MorphismFromTensorProductToInternalHomWithGivenObjects

  7.4-44 AddMorphismFromTensorProductToInternalHomWithGivenObjects

  7.4-45 IsomorphismFromTensorProductToInternalHom

  7.4-46 AddIsomorphismFromTensorProductToInternalHom

  7.4-47 MorphismFromInternalHomToTensorProduct

  7.4-48 MorphismFromInternalHomToTensorProductWithGivenObjects

  7.4-49 AddMorphismFromInternalHomToTensorProductWithGivenObjects

  7.4-50 IsomorphismFromInternalHomToTensorProduct

  7.4-51 AddIsomorphismFromInternalHomToTensorProduct

  7.4-52 TraceMap

  7.4-53 AddTraceMap

  7.4-54 RankMorphism

  7.4-55 AddRankMorphism

  7.4-56 IsomorphismFromDualToInternalHom

  7.4-57 AddIsomorphismFromDualToInternalHom

  7.4-58 IsomorphismFromInternalHomToDual

  7.4-59 AddIsomorphismFromInternalHomToDual

  7.4-60 UniversalPropertyOfDual

  7.4-61 AddUniversalPropertyOfDual

  7.4-62 LambdaIntroduction

  7.4-63 AddLambdaIntroduction

  7.4-64 LambdaElimination

  7.4-65 AddLambdaElimination

  7.4-66 IsomorphismFromObjectToInternalHom

  7.4-67 IsomorphismFromObjectToInternalHomWithGivenInternalHom

  7.4-68 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

  7.4-69 IsomorphismFromInternalHomToObject

  7.4-70 IsomorphismFromInternalHomToObjectWithGivenInternalHom

  7.4-71 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
 7.5 Rigid Symmetric Closed Monoidal Categories

7 Tensor Product and Internal Hom

7.1 Monoidal Categories

A \(6\)-tuple \(( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho )\) consisting of

is called a monoidal category, if

The corresponding GAP property is given by IsMonoidalCategory.

7.1-1 TensorProductOnObjects
‣ TensorProductOnObjects( a, b )( operation )

Returns: an object

The arguments are two objects \(a, b\). The output is the tensor product \(a \otimes b\).

7.1-2 AddTensorProductOnObjects
‣ AddTensorProductOnObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductOnObjects. \(F: (a,b) \mapsto a \otimes b\).

7.1-3 TensorProductOnMorphisms
‣ TensorProductOnMorphisms( alpha, beta )( operation )

Returns: a morphism in \(\mathrm{Hom}(a \otimes b, a' \otimes b')\)

The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the tensor product \(\alpha \otimes \beta\).

7.1-4 TensorProductOnMorphismsWithGivenTensorProducts
‣ TensorProductOnMorphismsWithGivenTensorProducts( s, alpha, beta, r )( operation )

Returns: a morphism in \(\mathrm{Hom}(a \otimes b, a' \otimes b')\)

The arguments are an object \(s = a \otimes b\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = a' \otimes b'\). The output is the tensor product \(\alpha \otimes \beta\).

7.1-5 AddTensorProductOnMorphismsWithGivenTensorProducts
‣ AddTensorProductOnMorphismsWithGivenTensorProducts( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductOnMorphismsWithGivenTensorProducts. \(F: ( a \otimes b, \alpha: a \rightarrow a', \beta: b \rightarrow b', a' \otimes b' ) \mapsto \alpha \otimes \beta\).

7.1-6 AssociatorRightToLeft
‣ AssociatorRightToLeft( a, b, c )( operation )

Returns: a morphism in \(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\).

The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\).

7.1-7 AssociatorRightToLeftWithGivenTensorProducts
‣ AssociatorRightToLeftWithGivenTensorProducts( s, a, b, c, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\).

The arguments are an object \(s = a \otimes (b \otimes c)\), three objects \(a,b,c\), and an object \(r = (a \otimes b) \otimes c\). The output is the associator \(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\).

7.1-8 AddAssociatorRightToLeftWithGivenTensorProducts
‣ AddAssociatorRightToLeftWithGivenTensorProducts( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation AssociatorRightToLeftWithGivenTensorProducts. \(F: ( a \otimes (b \otimes c), a, b, c, (a \otimes b) \otimes c ) \mapsto \alpha_{a,(b,c)}\).

7.1-9 AssociatorLeftToRight
‣ AssociatorLeftToRight( a, b, c )( operation )

Returns: a morphism in \(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\).

The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\).

7.1-10 AssociatorLeftToRightWithGivenTensorProducts
‣ AssociatorLeftToRightWithGivenTensorProducts( s, a, b, c, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\).

The arguments are an object \(s = (a \otimes b) \otimes c\), three objects \(a,b,c\), and an object \(r = a \otimes (b \otimes c)\). The output is the associator \(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\).

7.1-11 AddAssociatorLeftToRightWithGivenTensorProducts
‣ AddAssociatorLeftToRightWithGivenTensorProducts( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation AssociatorLeftToRightWithGivenTensorProducts. \(F: (( a \otimes b ) \otimes c, a, b, c, a \otimes (b \otimes c )) \mapsto \alpha_{(a,b),c}\).

7.1-12 TensorUnit
‣ TensorUnit( C )( attribute )

Returns: an object

The argument is a category \(\mathbf{C}\). The output is the tensor unit \(1\) of \(\mathbf{C}\).

7.1-13 AddTensorUnit
‣ AddTensorUnit( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorUnit. \(F: ( ) \mapsto 1\).

7.1-14 LeftUnitor
‣ LeftUnitor( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(1 \otimes a, a )\)

The argument is an object \(a\). The output is the left unitor \(\lambda_a: 1 \otimes a \rightarrow a\).

7.1-15 LeftUnitorWithGivenTensorProduct
‣ LeftUnitorWithGivenTensorProduct( a, s )( operation )

Returns: a morphism in \(\mathrm{Hom}(1 \otimes a, a )\)

The arguments are an object \(a\) and an object \(s = 1 \otimes a\). The output is the left unitor \(\lambda_a: 1 \otimes a \rightarrow a\).

7.1-16 AddLeftUnitorWithGivenTensorProduct
‣ AddLeftUnitorWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LeftUnitorWithGivenTensorProduct. \(F: (a, 1 \otimes a) \mapsto \lambda_a\).

7.1-17 LeftUnitorInverse
‣ LeftUnitorInverse( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(a, 1 \otimes a)\)

The argument is an object \(a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \otimes a\).

7.1-18 LeftUnitorInverseWithGivenTensorProduct
‣ LeftUnitorInverseWithGivenTensorProduct( a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}(a, 1 \otimes a)\)

The argument is an object \(a\) and an object \(r = 1 \otimes a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \otimes a\).

7.1-19 AddLeftUnitorInverseWithGivenTensorProduct
‣ AddLeftUnitorInverseWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LeftUnitorInverseWithGivenTensorProduct. \(F: (a, 1 \otimes a) \mapsto \lambda_a^{-1}\).

7.1-20 RightUnitor
‣ RightUnitor( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(a \otimes 1, a )\)

The argument is an object \(a\). The output is the right unitor \(\rho_a: a \otimes 1 \rightarrow a\).

7.1-21 RightUnitorWithGivenTensorProduct
‣ RightUnitorWithGivenTensorProduct( a, s )( operation )

Returns: a morphism in \(\mathrm{Hom}(a \otimes 1, a )\)

The arguments are an object \(a\) and an object \(s = a \otimes 1\). The output is the right unitor \(\rho_a: a \otimes 1 \rightarrow a\).

7.1-22 AddRightUnitorWithGivenTensorProduct
‣ AddRightUnitorWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation RightUnitorWithGivenTensorProduct. \(F: (a, a \otimes 1) \mapsto \rho_a\).

7.1-23 RightUnitorInverse
‣ RightUnitorInverse( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}( a, a \otimes 1 )\)

The argument is an object \(a\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \otimes 1\).

7.1-24 RightUnitorInverseWithGivenTensorProduct
‣ RightUnitorInverseWithGivenTensorProduct( a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a, a \otimes 1 )\)

The arguments are an object \(a\) and an object \(r = a \otimes 1\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \otimes 1\).

7.1-25 AddRightUnitorInverseWithGivenTensorProduct
‣ AddRightUnitorInverseWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation RightUnitorInverseWithGivenTensorProduct. \(F: (a, a \otimes 1) \mapsto \rho_a^{-1}\).

7.1-26 LeftDistributivityExpanding
‣ LeftDistributivityExpanding( a, L )( operation )

Returns: a morphism in \(\mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )\)

The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \(a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\).

7.1-27 LeftDistributivityExpandingWithGivenObjects
‣ LeftDistributivityExpandingWithGivenObjects( s, a, L, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( s, r )\)

The arguments are an object \(s = a \otimes (b_1 \oplus \dots \oplus b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).

7.1-28 AddLeftDistributivityExpandingWithGivenObjects
‣ AddLeftDistributivityExpandingWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LeftDistributivityExpandingWithGivenObjects. \(F: (a \otimes (b_1 \oplus \dots \oplus b_n), a, L, (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)) \mapsto \mathrm{LeftDistributivityExpandingWithGivenObjects}(a,L)\).

7.1-29 LeftDistributivityFactoring
‣ LeftDistributivityFactoring( a, L )( operation )

Returns: a morphism in \(\mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )\)

The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \((a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n)\).

7.1-30 LeftDistributivityFactoringWithGivenObjects
‣ LeftDistributivityFactoringWithGivenObjects( s, a, L, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( s, r )\)

The arguments are an object \(s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = a \otimes (b_1 \oplus \dots \oplus b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).

7.1-31 AddLeftDistributivityFactoringWithGivenObjects
‣ AddLeftDistributivityFactoringWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LeftDistributivityFactoringWithGivenObjects. \(F: ((a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a, L, a \otimes (b_1 \oplus \dots \oplus b_n)) \mapsto \mathrm{LeftDistributivityFactoringWithGivenObjects}(a,L)\).

7.1-32 RightDistributivityExpanding
‣ RightDistributivityExpanding( L, a )( operation )

Returns: a morphism in \(\mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )\)

The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\).

7.1-33 RightDistributivityExpandingWithGivenObjects
‣ RightDistributivityExpandingWithGivenObjects( s, L, a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( s, r )\)

The arguments are an object \(s = (b_1 \oplus \dots \oplus b_n) \otimes a\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\). The output is the right distributivity morphism \(s \rightarrow r\).

7.1-34 AddRightDistributivityExpandingWithGivenObjects
‣ AddRightDistributivityExpandingWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation RightDistributivityExpandingWithGivenObjects. \(F: ((b_1 \oplus \dots \oplus b_n) \otimes a, L, a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)) \mapsto \mathrm{RightDistributivityExpandingWithGivenObjects}(L,a)\).

7.1-35 RightDistributivityFactoring
‣ RightDistributivityFactoring( L, a )( operation )

Returns: a morphism in \(\mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)\)

The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a \).

7.1-36 RightDistributivityFactoringWithGivenObjects
‣ RightDistributivityFactoringWithGivenObjects( s, L, a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( s, r )\)

The arguments are an object \(s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \oplus \dots \oplus b_n) \otimes a\). The output is the right distributivity morphism \(s \rightarrow r\).

7.1-37 AddRightDistributivityFactoringWithGivenObjects
‣ AddRightDistributivityFactoringWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation RightDistributivityFactoringWithGivenObjects. \(F: ((b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), L, a, (b_1 \oplus \dots \oplus b_n) \otimes a) \mapsto \mathrm{RightDistributivityFactoringWithGivenObjects}(L,a)\).

7.2 Braided Monoidal Categories

A monoidal category \(\mathbf{C}\) equipped with a natural isomorphism \(B_{a,b}: a \otimes b \cong b \otimes a\) is called a braided monoidal category if

The corresponding GAP property is given by IsBraidedMonoidalCategory.

7.2-1 Braiding
‣ Braiding( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( a \otimes b, b \otimes a )\).

The arguments are two objects \(a,b\). The output is the braiding \( B_{a,b}: a \otimes b \rightarrow b \otimes a\).

7.2-2 BraidingWithGivenTensorProducts
‣ BraidingWithGivenTensorProducts( s, a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a \otimes b, b \otimes a )\).

The arguments are an object \(s = a \otimes b\), two objects \(a,b\), and an object \(r = b \otimes a\). The output is the braiding \( B_{a,b}: a \otimes b \rightarrow b \otimes a\).

7.2-3 AddBraidingWithGivenTensorProducts
‣ AddBraidingWithGivenTensorProducts( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation BraidingWithGivenTensorProducts. \(F: (a \otimes b, a, b, b \otimes a) \rightarrow B_{a,b}\).

7.2-4 BraidingInverse
‣ BraidingInverse( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( b \otimes a, a \otimes b )\).

The arguments are two objects \(a,b\). The output is the inverse of the braiding \( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\).

7.2-5 BraidingInverseWithGivenTensorProducts
‣ BraidingInverseWithGivenTensorProducts( s, a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( b \otimes a, a \otimes b )\).

The arguments are an object \(s = b \otimes a\), two objects \(a,b\), and an object \(r = a \otimes b\). The output is the braiding \( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\).

7.2-6 AddBraidingInverseWithGivenTensorProducts
‣ AddBraidingInverseWithGivenTensorProducts( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation BraidingInverseWithGivenTensorProducts. \(F: (b \otimes a, a, b, a \otimes b) \rightarrow B_{a,b}^{-1}\).

7.3 Symmetric Monoidal Categories

A braided monoidal category \(\mathbf{C}\) is called symmetric monoidal category if \(B_{a,b}^{-1} = B_{b,a}\). The corresponding GAP property is given by IsSymmetricMonoidalCategory.

7.4 Symmetric Closed Monoidal Categories

A symmetric monoidal category \(\mathbf{C}\) which has for each functor \(- \otimes b: \mathbf{C} \rightarrow \mathbf{C}\) a right adjoint (denoted by \(\mathrm{\underline{Hom}}(b,-)\)) is called a symmetric closed monoidal category. The corresponding GAP property is given by IsSymmetricClosedMonoidalCategory.

7.4-1 InternalHomOnObjects
‣ InternalHomOnObjects( a, b )( operation )

Returns: an object

The arguments are two objects \(a,b\). The output is the internal hom object \(\mathrm{\underline{Hom}}(a,b)\).

7.4-2 AddInternalHomOnObjects
‣ AddInternalHomOnObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation InternalHomOnObjects. \(F: (a,b) \mapsto \mathrm{\underline{Hom}}(a,b)\).

7.4-3 InternalHomOnMorphisms
‣ InternalHomOnMorphisms( alpha, beta )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )\)

The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the internal hom morphism \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).

7.4-4 InternalHomOnMorphismsWithGivenInternalHoms
‣ InternalHomOnMorphismsWithGivenInternalHoms( s, alpha, beta, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )\)

The arguments are an object \(s = \mathrm{\underline{Hom}}(a',b)\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = \mathrm{\underline{Hom}}(a,b')\). The output is the internal hom morphism \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).

7.4-5 AddInternalHomOnMorphismsWithGivenInternalHoms
‣ AddInternalHomOnMorphismsWithGivenInternalHoms( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation InternalHomOnMorphismsWithGivenInternalHoms. \(F: (\mathrm{\underline{Hom}}(a',b), \alpha: a \rightarrow a', \beta: b \rightarrow b', \mathrm{\underline{Hom}}(a,b') ) \mapsto \mathrm{\underline{Hom}}(\alpha,\beta)\).

7.4-6 EvaluationMorphism
‣ EvaluationMorphism( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )\).

The arguments are two objects \(a, b\). The output is the evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b\), i.e., the counit of the tensor hom adjunction.

7.4-7 EvaluationMorphismWithGivenSource
‣ EvaluationMorphismWithGivenSource( a, b, s )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )\).

The arguments are two objects \(a,b\) and an object \(s = \mathrm{\underline{Hom}}(a,b) \otimes a\). The output is the evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b\), i.e., the counit of the tensor hom adjunction.

7.4-8 AddEvaluationMorphismWithGivenSource
‣ AddEvaluationMorphismWithGivenSource( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation EvaluationMorphismWithGivenSource. \(F: (a, b, \mathrm{\underline{Hom}}(a,b) \otimes a) \mapsto \mathrm{ev}_{a,b}\).

7.4-9 CoevaluationMorphism
‣ CoevaluationMorphism( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )\).

The arguments are two objects \(a,b\). The output is the coevaluation morphism \(\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}(b, a \otimes b)}\), i.e., the unit of the tensor hom adjunction.

7.4-10 CoevaluationMorphismWithGivenRange
‣ CoevaluationMorphismWithGivenRange( a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )\).

The arguments are two objects \(a,b\) and an object \(r = \mathrm{\underline{Hom}(b, a \otimes b)}\). The output is the coevaluation morphism \(\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}(b, a \otimes b)}\), i.e., the unit of the tensor hom adjunction.

7.4-11 AddCoevaluationMorphismWithGivenRange
‣ AddCoevaluationMorphismWithGivenRange( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation CoevaluationMorphismWithGivenRange. \(F: (a, b, \mathrm{\underline{Hom}}(b, a \otimes b)) \mapsto \mathrm{coev}_{a,b}\).

7.4-12 TensorProductToInternalHomAdjunctionMap
‣ TensorProductToInternalHomAdjunctionMap( a, b, f )( operation )

Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) )\).

The arguments are objects \(a,b\) and a morphism \(f: a \otimes b \rightarrow c\). The output is a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\) corresponding to \(f\) under the tensor hom adjunction.

7.4-13 AddTensorProductToInternalHomAdjunctionMap
‣ AddTensorProductToInternalHomAdjunctionMap( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductToInternalHomAdjunctionMap. \(F: (a, b, f: a \otimes b \rightarrow c) \mapsto ( g: a \rightarrow \mathrm{\underline{Hom}}(b,c) )\).

7.4-14 InternalHomToTensorProductAdjunctionMap
‣ InternalHomToTensorProductAdjunctionMap( b, c, g )( operation )

Returns: a morphism in \(\mathrm{Hom}(a \otimes b, c)\).

The arguments are objects \(b,c\) and a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\). The output is a morphism \(f: a \otimes b \rightarrow c\) corresponding to \(g\) under the tensor hom adjunction.

7.4-15 AddInternalHomToTensorProductAdjunctionMap
‣ AddInternalHomToTensorProductAdjunctionMap( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation InternalHomToTensorProductAdjunctionMap. \(F: (b, c, g: a \rightarrow \mathrm{\underline{Hom}}(b,c)) \mapsto ( g: a \otimes b \rightarrow c )\).

7.4-16 MonoidalPreComposeMorphism
‣ MonoidalPreComposeMorphism( a, b, c )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )\).

The arguments are three objects \(a,b,c\). The output is the precomposition morphism \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

7.4-17 MonoidalPreComposeMorphismWithGivenObjects
‣ MonoidalPreComposeMorphismWithGivenObjects( s, a, b, c, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )\).

The arguments are an object \(s = \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{Hom}}(a,c)\). The output is the precomposition morphism \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

7.4-18 AddMonoidalPreComposeMorphismWithGivenObjects
‣ AddMonoidalPreComposeMorphismWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MonoidalPreComposeMorphismWithGivenObjects. \(F: (\mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c),a,b,c,\mathrm{\underline{Hom}}(a,c)) \mapsto \mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}\).

7.4-19 MonoidalPostComposeMorphism
‣ MonoidalPostComposeMorphism( a, b, c )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )\).

The arguments are three objects \(a,b,c\). The output is the postcomposition morphism \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

7.4-20 MonoidalPostComposeMorphismWithGivenObjects
‣ MonoidalPostComposeMorphismWithGivenObjects( s, a, b, c, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )\).

The arguments are an object \(s = \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{Hom}}(a,c)\). The output is the postcomposition morphism \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

7.4-21 AddMonoidalPostComposeMorphismWithGivenObjects
‣ AddMonoidalPostComposeMorphismWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MonoidalPostComposeMorphismWithGivenObjects. \(F: (\mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b),a,b,c,\mathrm{\underline{Hom}}(a,c)) \mapsto \mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}\).

7.4-22 DualOnObjects
‣ DualOnObjects( a )( attribute )

Returns: an object

The argument is an object \(a\). The output is its dual object \(a^{\vee}\).

7.4-23 AddDualOnObjects
‣ AddDualOnObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation DualOnObjects. \(F: a \mapsto a^{\vee}\).

7.4-24 DualOnMorphisms
‣ DualOnMorphisms( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}( b^{\vee}, a^{\vee} )\).

The argument is a morphism \(\alpha: a \rightarrow b\). The output is its dual morphism \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).

7.4-25 DualOnMorphismsWithGivenDuals
‣ DualOnMorphismsWithGivenDuals( s, alpha, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( b^{\vee}, a^{\vee} )\).

The argument is an object \(s = b^{\vee}\), a morphism \(\alpha: a \rightarrow b\), and an object \(r = a^{\vee}\). The output is the dual morphism \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).

7.4-26 AddDualOnMorphismsWithGivenDuals
‣ AddDualOnMorphismsWithGivenDuals( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation DualOnMorphismsWithGivenDuals. \(F: (b^{\vee},\alpha,a^{\vee}) \mapsto \alpha^{\vee}\).

7.4-27 EvaluationForDual
‣ EvaluationForDual( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes a, 1 )\).

The argument is an object \(a\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\).

7.4-28 EvaluationForDualWithGivenTensorProduct
‣ EvaluationForDualWithGivenTensorProduct( s, a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes a, 1 )\).

The arguments are an object \(s = a^{\vee} \otimes a\), an object \(a\), and an object \(r = 1\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\).

7.4-29 AddEvaluationForDualWithGivenTensorProduct
‣ AddEvaluationForDualWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation EvaluationForDualWithGivenTensorProduct. \(F: (a^{\vee} \otimes a, a, 1) \mapsto \mathrm{ev}_{a}\).

7.4-30 CoevaluationForDual
‣ CoevaluationForDual( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(1,a \otimes a^{\vee})\).

The argument is an object \(a\). The output is the coevaluation morphism \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).

7.4-31 CoevaluationForDualWithGivenTensorProduct
‣ CoevaluationForDualWithGivenTensorProduct( s, a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}(1,a \otimes a^{\vee})\).

The arguments are an object \(s = 1\), an object \(a\), and an object \(r = a \otimes a^{\vee}\). The output is the coevaluation morphism \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).

7.4-32 AddCoevaluationForDualWithGivenTensorProduct
‣ AddCoevaluationForDualWithGivenTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation CoevaluationForDualWithGivenTensorProduct. \(F: (1, a, a \otimes a^{\vee}) \mapsto \mathrm{coev}_{a}\).

7.4-33 MorphismToBidual
‣ MorphismToBidual( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(a, (a^{\vee})^{\vee})\).

The argument is an object \(a\). The output is the morphism to the bidual \(a \rightarrow (a^{\vee})^{\vee}\).

7.4-34 MorphismToBidualWithGivenBidual
‣ MorphismToBidualWithGivenBidual( a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}(a, (a^{\vee})^{\vee})\).

The arguments are an object \(a\), and an object \(r = (a^{\vee})^{\vee}\). The output is the morphism to the bidual \(a \rightarrow (a^{\vee})^{\vee}\).

7.4-35 AddMorphismToBidualWithGivenBidual
‣ AddMorphismToBidualWithGivenBidual( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MorphismToBidualWithGivenBidual. \(F: (a, (a^{\vee})^{\vee}) \mapsto (a \rightarrow (a^{\vee})^{\vee})\).

7.4-36 TensorProductInternalHomCompatibilityMorphism
‣ TensorProductInternalHomCompatibilityMorphism( a, a', b, b' )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))\).

The arguments are four objects \(a, a', b, b'\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).

7.4-37 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
‣ TensorProductInternalHomCompatibilityMorphismWithGivenObjects( a, a', b, b', L )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))\).

The arguments are four objects \(a, a', b, b'\), and a list \(L = [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).

7.4-38 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
‣ AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphismWithGivenObjects. \(F: ( a,a',b,b', [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]) \mapsto \mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}\).

7.4-39 TensorProductDualityCompatibilityMorphism
‣ TensorProductDualityCompatibilityMorphism( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )\).

The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).

7.4-40 TensorProductDualityCompatibilityMorphismWithGivenObjects
‣ TensorProductDualityCompatibilityMorphismWithGivenObjects( s, a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )\).

The arguments are an object \(s = a^{\vee} \otimes b^{\vee}\), two objects \(a,b\), and an object \(r = (a \otimes b)^{\vee}\). The output is the natural morphism \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).

7.4-41 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
‣ AddTensorProductDualityCompatibilityMorphismWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductDualityCompatibilityMorphismWithGivenObjects. \(F: ( a^{\vee} \otimes b^{\vee}, a, b, (a \otimes b)^{\vee} ) \mapsto \mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}\).

7.4-42 MorphismFromTensorProductToInternalHom
‣ MorphismFromTensorProductToInternalHom( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).

The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

7.4-43 MorphismFromTensorProductToInternalHomWithGivenObjects
‣ MorphismFromTensorProductToInternalHomWithGivenObjects( s, a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).

The arguments are an object \(s = a^{\vee} \otimes b\), two objects \(a,b\), and an object \(r = \mathrm{\underline{Hom}}(a,b)\). The output is the natural morphism \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

7.4-44 AddMorphismFromTensorProductToInternalHomWithGivenObjects
‣ AddMorphismFromTensorProductToInternalHomWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MorphismFromTensorProductToInternalHomWithGivenObjects. \(F: ( a^{\vee} \otimes b, a, b, \mathrm{\underline{Hom}}(a,b) ) \mapsto \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}\).

7.4-45 IsomorphismFromTensorProductToInternalHom
‣ IsomorphismFromTensorProductToInternalHom( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).

The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{IsomorphismFromTensorProductToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

7.4-46 AddIsomorphismFromTensorProductToInternalHom
‣ AddIsomorphismFromTensorProductToInternalHom( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromTensorProductToInternalHom. \(F: ( a, b ) \mapsto \mathrm{IsomorphismFromTensorProductToInternalHom}_{a,b}\).

7.4-47 MorphismFromInternalHomToTensorProduct
‣ MorphismFromInternalHomToTensorProduct( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).

The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}\), namely \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

7.4-48 MorphismFromInternalHomToTensorProductWithGivenObjects
‣ MorphismFromInternalHomToTensorProductWithGivenObjects( s, a, b, r )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).

The arguments are an object \(s = \mathrm{\underline{Hom}}(a,b)\), two objects \(a,b\), and an object \(r = a^{\vee} \otimes b\). The output is the inverse of \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}\), namely \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

7.4-49 AddMorphismFromInternalHomToTensorProductWithGivenObjects
‣ AddMorphismFromInternalHomToTensorProductWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MorphismFromInternalHomToTensorProductWithGivenObjects. \(F: ( \mathrm{\underline{Hom}}(a,b),a,b,a^{\vee} \otimes b ) \mapsto \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}\).

7.4-50 IsomorphismFromInternalHomToTensorProduct
‣ IsomorphismFromInternalHomToTensorProduct( a, b )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).

The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{IsomorphismFromTensorProductToInternalHom}\), namely \(\mathrm{IsomorphismFromInternalHomToTensorProduct}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

7.4-51 AddIsomorphismFromInternalHomToTensorProduct
‣ AddIsomorphismFromInternalHomToTensorProduct( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToTensorProduct. \(F: ( a,b ) \mapsto \mathrm{IsomorphismFromInternalHomToTensorProduct}_{a,b}\).

7.4-52 TraceMap
‣ TraceMap( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}(1,1)\).

The argument is a morphism \(\alpha\). The output is the trace morphism \(\mathrm{trace}_{\alpha}: 1 \rightarrow 1\).

7.4-53 AddTraceMap
‣ AddTraceMap( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TraceMap. \(F: \alpha \mapsto \mathrm{trace}_{\alpha}\)

7.4-54 RankMorphism
‣ RankMorphism( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(1,1)\).

The argument is an object \(a\). The output is the rank morphism \(\mathrm{rank}_a: 1 \rightarrow 1\).

7.4-55 AddRankMorphism
‣ AddRankMorphism( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation RankMorphism. \(F: a \mapsto \mathrm{rank}_{a}\)

7.4-56 IsomorphismFromDualToInternalHom
‣ IsomorphismFromDualToInternalHom( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}}(a,1))\).

The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromDualToInternalHom}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1)\).

7.4-57 AddIsomorphismFromDualToInternalHom
‣ AddIsomorphismFromDualToInternalHom( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromDualToInternalHom. \(F: a \mapsto \mathrm{IsomorphismFromDualToInternalHom}_{a}\)

7.4-58 IsomorphismFromInternalHomToDual
‣ IsomorphismFromInternalHomToDual( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(a,1), a^{\vee})\).

The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromInternalHomToDual}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}\).

7.4-59 AddIsomorphismFromInternalHomToDual
‣ AddIsomorphismFromInternalHomToDual( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToDual. \(F: a \mapsto \mathrm{IsomorphismFromInternalHomToDual}_{a}\)

7.4-60 UniversalPropertyOfDual
‣ UniversalPropertyOfDual( t, a, alpha )( operation )

Returns: a morphism in \(\mathrm{Hom}(t, a^{\vee})\).

The arguments are two objects \(t,a\), and a morphism \(\alpha: t \otimes a \rightarrow 1\). The output is the morphism \(t \rightarrow a^{\vee}\) given by the universal property of \(a^{\vee}\).

7.4-61 AddUniversalPropertyOfDual
‣ AddUniversalPropertyOfDual( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation UniversalPropertyOfDual. \(F: ( t,a,\alpha: t \otimes a \rightarrow 1 ) \mapsto ( t \rightarrow a^{\vee} )\).

7.4-62 LambdaIntroduction
‣ LambdaIntroduction( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) )\).

The argument is a morphism \(\alpha: a \rightarrow b\). The output is the corresponding morphism \(1 \rightarrow \mathrm{\underline{Hom}}(a,b)\) under the tensor hom adjunction.

7.4-63 AddLambdaIntroduction
‣ AddLambdaIntroduction( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LambdaIntroduction. \(F: ( \alpha: a \rightarrow b ) \mapsto ( 1 \rightarrow \mathrm{\underline{Hom}}(a,b) )\).

7.4-64 LambdaElimination
‣ LambdaElimination( a, b, alpha )( operation )

Returns: a morphism in \(\mathrm{Hom}(a,b)\).

The arguments are two objects \(a,b\), and a morphism \(\alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b)\). The output is a morphism \(a \rightarrow b\) corresponding to \(\alpha\) under the tensor hom adjunction.

7.4-65 AddLambdaElimination
‣ AddLambdaElimination( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LambdaElimination. \(F: ( a,b,\alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b) ) \mapsto ( a \rightarrow b )\).

7.4-66 IsomorphismFromObjectToInternalHom
‣ IsomorphismFromObjectToInternalHom( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))\).

The argument is an object \(a\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{Hom}}(1,a)\).

7.4-67 IsomorphismFromObjectToInternalHomWithGivenInternalHom
‣ IsomorphismFromObjectToInternalHomWithGivenInternalHom( a, r )( operation )

Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))\).

The argument is an object \(a\), and an object \(r = \mathrm{\underline{Hom}}(1,a)\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{Hom}}(1,a)\).

7.4-68 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
‣ AddIsomorphismFromObjectToInternalHomWithGivenInternalHom( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToInternalHomWithGivenInternalHom. \(F: ( a, \mathrm{\underline{Hom}}(1,a) ) \mapsto ( a \rightarrow \mathrm{\underline{Hom}}(1,a) )\).

7.4-69 IsomorphismFromInternalHomToObject
‣ IsomorphismFromInternalHomToObject( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)\).

The argument is an object \(a\). The output is the natural isomorphism \(\mathrm{\underline{Hom}}(1,a) \rightarrow a\).

7.4-70 IsomorphismFromInternalHomToObjectWithGivenInternalHom
‣ IsomorphismFromInternalHomToObjectWithGivenInternalHom( a, s )( operation )

Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)\).

The argument is an object \(a\), and an object \(s = \mathrm{\underline{Hom}}(1,a)\). The output is the natural isomorphism \(\mathrm{\underline{Hom}}(1,a) \rightarrow a\).

7.4-71 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
‣ AddIsomorphismFromInternalHomToObjectWithGivenInternalHom( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToObjectWithGivenInternalHom. \(F: ( a, \mathrm{\underline{Hom}}(1,a) ) \mapsto ( \mathrm{\underline{Hom}}(1,a) \rightarrow a )\).

7.5 Rigid Symmetric Closed Monoidal Categories

A symmetric closed monoidal category \(\mathbf{C}\) satisfying

is called a rigid symmetric closed monoidal category.

7.5-1 TensorProductInternalHomCompatibilityMorphismInverse
‣ TensorProductInternalHomCompatibilityMorphismInverse( a, a', b, b' )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'))\).

The arguments are four objects \(a, a', b, b'\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).

7.5-2 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
‣ TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects( a, a', b, b', L )( operation )

Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'))\).

The arguments are four objects \(a, a', b, b'\), and a list \(L = [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).

7.5-3 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
‣ AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects. \(F: ( a,a',b,b', [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]) \mapsto \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}\).

7.5-4 MorphismFromBidual
‣ MorphismFromBidual( a )( attribute )

Returns: a morphism in \(\mathrm{Hom}((a^{\vee})^{\vee},a)\).

The argument is an object \(a\). The output is the inverse of the morphism to the bidual \((a^{\vee})^{\vee} \rightarrow a\).

7.5-5 MorphismFromBidualWithGivenBidual
‣ MorphismFromBidualWithGivenBidual( a, s )( operation )

Returns: a morphism in \(\mathrm{Hom}((a^{\vee})^{\vee},a)\).

The argument is an object \(a\), and an object \(s = (a^{\vee})^{\vee}\). The output is the inverse of the morphism to the bidual \((a^{\vee})^{\vee} \rightarrow a\).

7.5-6 AddMorphismFromBidualWithGivenBidual
‣ AddMorphismFromBidualWithGivenBidual( C, F )( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation MorphismFromBidualWithGivenBidual. \(F: (a, (a^{\vee})^{\vee}) \mapsto ((a^{\vee})^{\vee} \rightarrow a)\).

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