A 6-tuple ( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho ) consisting of
a category \mathbf{C},
a functor \otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C},
an object 1 \in \mathbf{C},
a natural isomorphism \alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c,
a natural isomorphism \lambda_{a}: 1 \otimes a \cong a,
a natural isomorphism \rho_{a}: a \otimes 1 \cong a,
is called a monoidal category, if
for all objects a,b,c,d, the pentagon identity holds: (\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) = \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d},
for all objects a,c, the triangle identity holds: ( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} = \mathrm{id}_a \otimes \lambda_c.
The corresponding GAP property is given by IsMonoidalCategory.
‣ TensorProductOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects a, b. The output is the tensor product a \otimes b.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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)}.
‣ 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).
‣ 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).
‣ 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}.
‣ TensorUnit( C ) | ( attribute ) |
Returns: an object
The argument is a category \mathbf{C}. The output is the tensor unit 1 of \mathbf{C}.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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}.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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}.
‣ 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).
‣ 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.
‣ 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).
‣ 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).
‣ 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.
‣ 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).
‣ 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).
‣ 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.
‣ 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).
‣ 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 .
‣ 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.
‣ 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).
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
\lambda_a \circ B_{a,1} = \rho_a,
(B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} = \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c},
( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} = \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}.
The corresponding GAP property is given by IsBraidedMonoidalCategory.
‣ 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.
‣ 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.
‣ 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}.
‣ 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.
‣ 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.
‣ 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}.
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.
A symmetric monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a right adjoint (denoted by \underline{\mathrm{Hom}}(b,-)) is called a symmetric closed monoidal category. The corresponding GAP property is given by IsSymmetricClosedMonoidalCategory.
‣ InternalHomOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects a,b. The output is the internal hom object \underline{\mathrm{Hom}}(a,b).
‣ 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 \underline{\mathrm{Hom}}(a,b).
‣ InternalHomOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \underline{\mathrm{Hom}}(a',b), \underline{\mathrm{Hom}}(a,b') )
The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal hom morphism \underline{\mathrm{Hom}}(\alpha,\beta): \underline{\mathrm{Hom}}(a',b) \rightarrow \underline{\mathrm{Hom}}(a,b').
‣ InternalHomOnMorphismsWithGivenInternalHoms( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \underline{\mathrm{Hom}}(a',b), \underline{\mathrm{Hom}}(a,b') )
The arguments are an object s = \underline{\mathrm{Hom}}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \underline{\mathrm{Hom}}(a,b'). The output is the internal hom morphism \underline{\mathrm{Hom}}(\alpha,\beta): \underline{\mathrm{Hom}}(a',b) \rightarrow \underline{\mathrm{Hom}}(a,b').
‣ 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: (\underline{\mathrm{Hom}}(a',b), \alpha: a \rightarrow a', \beta: b \rightarrow b', \underline{\mathrm{Hom}}(a,b') ) \mapsto \underline{\mathrm{Hom}}(\alpha,\beta).
‣ 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.
‣ 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.
‣ 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}.
‣ 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.
‣ 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.
‣ 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}.
‣ 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.
‣ 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) ).
‣ 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.
‣ 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 ).
‣ 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).
‣ 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).
‣ 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}.
‣ 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).
‣ 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).
‣ 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}.
‣ DualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object a. The output is its dual object a^{\vee}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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.
‣ 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.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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}).
‣ 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').
‣ 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').
‣ 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'}.
‣ 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}.
‣ 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}.
‣ 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}.
‣ 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).
‣ 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).
‣ 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}.
‣ 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).
‣ 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}.
‣ 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.
‣ 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.
‣ 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}.
‣ 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.
‣ 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}.
‣ 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.
‣ 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}
‣ 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.
‣ 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}
‣ IsomorphismFromDualToInternalHom( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{Hom}(a,1)).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromDualToInternalHom}_{a}: a^{\vee} \rightarrow \mathrm{Hom}(a,1).
‣ 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}
‣ IsomorphismFromInternalHomToDual( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{Hom}(a,1), a^{\vee}).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalHomToDual}_{a}: \mathrm{Hom}(a,1) \rightarrow a^{\vee}.
‣ 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}
‣ 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}.
‣ 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} ).
‣ 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.
‣ 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) ).
‣ 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.
‣ 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 ).
‣ 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).
‣ 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).
‣ 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) ).
‣ 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.
‣ 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.
‣ 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 ).
A symmetric closed monoidal category \mathbf{C} satisfying
the natural morphism \underline{\mathrm{Hom}}(a_1,b_1) \otimes \underline{\mathrm{Hom}}(a_2,b_2) \rightarrow \underline{\mathrm{Hom}}(a_1 \otimes a_2,b_1 \otimes b_2) is an isomorphism,
the natural morphism a \rightarrow \underline{\mathrm{Hom}}(\underline{\mathrm{Hom}}(a, 1), 1) is an isomorphism
is called a rigid symmetric closed monoidal category.
‣ 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').
‣ 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').
‣ 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'}.
‣ 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.
‣ 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.
‣ 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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