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} compatible with the congruence of morphisms,
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} ) \sim \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} \sim \mathrm{id}_a \otimes \lambda_c.
The corresponding GAP property is given by IsMonoidalCategory
.
‣ 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.
‣ 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.
‣ 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).
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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 operation adds the given function F to the category for the basic operation TensorProductOnObjects
. F: (a,b) \mapsto a \otimes b.
‣ 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 operation adds the given function F to the category for the basic operation TensorUnit
. F: ( ) \mapsto 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.
‣ 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.
‣ 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.
‣ 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.
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} \sim \rho_a,
(B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} \sim \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} \sim \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.
‣ 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 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 inverse braiding B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b.
A braided monoidal category \mathbf{C} is called symmetric monoidal category if B_{a,b}^{-1} \sim B_{b,a}. The corresponding GAP property is given by IsSymmetricMonoidalCategory
.
A 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 closed monoidal category.
If no operations involving duals are installed manually, the dual objects will be derived as a^\vee \coloneqq \mathrm{\underline{Hom}}(a,1).
The corresponding GAP property is called IsClosedMonoidalCategory
.
‣ 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).
‣ 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').
‣ 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').
‣ 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.
‣ 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.
‣ TensorProductToInternalHomAdjunctionMap ( a, b, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) ).
The arguments are two 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.
‣ TensorProductToInternalHomAdjunctionMapWithGivenInternalHom ( a, b, f, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) ).
The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}}(b,c). The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) corresponding to f under the tensor hom adjunction.
‣ InternalHomToTensorProductAdjunctionMap ( b, c, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a \otimes b, c).
The arguments are two 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.
‣ InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct ( b, c, g, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a \otimes b, c).
The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) and an object t = a \otimes b. The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.
‣ 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).
‣ 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).
‣ DualOnObjects ( a ) | ( attribute ) |
Returns: an object
The argument is an object a. The output is its dual object 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}.
‣ 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.
‣ 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}.
‣ TensorProductInternalHomCompatibilityMorphism ( list ) | ( 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 argument is a list of 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 ( s, list, r ) | ( 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 a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') and r = \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').
‣ 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}.
‣ 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).
‣ IsomorphismFromDualObjectToInternalHomIntoTensorUnit ( 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{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1).
‣ IsomorphismFromInternalHomIntoTensorUnitToDualObject ( 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{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}.
‣ 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}.
‣ 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.
‣ 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.
‣ 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).
‣ 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.
A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a left adjoint (denoted by \mathrm{\underline{coHom}}(-,b)) is called a coclosed monoidal category.
If no operations involving coduals are installed manually, the codual objects will be derived as a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a).
The corresponding GAP property is called IsCoclosedMonoidalCategory
.
‣ InternalCoHomOnObjects ( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}}(a,b).
‣ InternalCoHomOnMorphisms ( alpha, beta ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )
The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).
‣ InternalCoHomOnMorphismsWithGivenInternalCoHoms ( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )
The arguments are an object s = \mathrm{\underline{coHom}}(a,b'), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{coHom}}(a',b). The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).
‣ CoclosedEvaluationMorphism ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b ).
The arguments are two objects a, b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b, i.e., the unit of the cohom tensor adjunction.
‣ CoclosedEvaluationMorphismWithGivenRange ( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b ).
The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}}(a,b) \otimes b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b, i.e., the unit of the cohom tensor adjunction.
‣ CoclosedCoevaluationMorphism ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), a ).
The arguments are two objects a,b. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a, i.e., the counit of the cohom tensor adjunction.
‣ CoclosedCoevaluationMorphismWithGivenSource ( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}(a \otimes b, b)}. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a, i.e., the unit of the cohom tensor adjunction.
‣ TensorProductToInternalCoHomAdjunctionMap ( c, b, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).
The arguments are two objects c,b and a morphism g: a \rightarrow c \otimes b. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.
‣ TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom ( c, b, g, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).
The arguments are two objects c,b, a morphism g: a \rightarrow c \otimes b and an object i = \mathrm{\underline{coHom}(a,b)}. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductAdjunctionMap ( a, b, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, c \otimes b).
The arguments are two objects a,b and a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c. The output is a morphism g: a \rightarrow c \otimes b corresponding to f under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct ( a, b, f, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, c \otimes b).
The arguments are two objects a,b, a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c and an object t = c \otimes b. The output is a morphism g: a \rightarrow c \otimes b corresponding to f under the cohom tensor adjunction.
‣ MonoidalPreCoComposeMorphism ( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) ).
The arguments are three objects a,b,c. The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).
‣ MonoidalPreCoComposeMorphismWithGivenObjects ( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) ).
The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c). The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).
‣ MonoidalPostCoComposeMorphism ( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) ).
The arguments are three objects a,b,c. The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).
‣ MonoidalPostCoComposeMorphismWithGivenObjects ( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) ).
The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b). The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).
‣ CoDualOnObjects ( a ) | ( attribute ) |
Returns: an object
The argument is an object a. The output is its codual object a_{\vee}.
‣ CoDualOnMorphisms ( 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 codual morphism \alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}.
‣ CoDualOnMorphismsWithGivenCoDuals ( 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}.
‣ CoclosedEvaluationForCoDual ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).
The argument is an object a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.
‣ CoclosedEvaluationForCoDualWithGivenTensorProduct ( s, a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).
The arguments are an object s = 1, an object a, and an object r = a_{\vee} \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.
‣ MorphismFromCoBidual ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).
The argument is an object a. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.
‣ MorphismFromCoBidualWithGivenCoBidual ( a, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).
The arguments are an object a, and an object s = (a_{\vee})_{\vee}. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.
‣ InternalCoHomTensorProductCompatibilityMorphism ( list ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')).
The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').
‣ InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects ( s, list, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') ).
The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') and r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').
‣ CoDualityTensorProductCompatibilityMorphism ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).
The arguments are two objects a,b. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.
‣ CoDualityTensorProductCompatibilityMorphismWithGivenObjects ( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).
The arguments are an object s = (a \otimes b)_{\vee}, two objects a,b, and an object r = a_{\vee} \otimes b_{\vee}. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.
‣ MorphismFromInternalCoHomToTensorProduct ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).
The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.
‣ MorphismFromInternalCoHomToTensorProductWithGivenObjects ( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), a \otimes b_{\vee} ).
The arguments are an object s = \mathrm{\underline{coHom}}(a,b), two objects a,b, and an object r = b_{\vee} \otimes a. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}.
‣ IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}}(1,a)).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a).
‣ IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(1,a), a_{\vee}).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}: \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}.
‣ UniversalPropertyOfCoDual ( t, a, alpha ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a_{\vee}, t).
The arguments are two objects t,a, and a morphism \alpha: 1 \rightarrow t \otimes a. The output is the morphism a_{\vee} \rightarrow t given by the universal property of a_{\vee}.
‣ CoLambdaIntroduction ( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), 1 ).
The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism \mathrm{\underline{coHom}}(a,b) \rightarrow 1 under the cohom tensor adjunction.
‣ CoLambdaElimination ( a, b, alpha ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a,b).
The arguments are two objects a,b, and a morphism \alpha: \mathrm{\underline{coHom}}(a,b) \rightarrow 1. The output is a morphism a \rightarrow b corresponding to \alpha under the cohom tensor adjunction.
‣ IsomorphismFromObjectToInternalCoHom ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1)).
The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).
‣ IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom ( a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1)).
The argument is an object a, and an object r = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).
‣ IsomorphismFromInternalCoHomToObject ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a).
The argument is an object a. The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.
‣ IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom ( a, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a).
The argument is an object a, and an object s = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.
A monoidal category \mathbf{C} which is symmetric and closed is called a symmetric closed monoidal category.
The corresponding GAP property is given by IsSymmetricClosedMonoidalCategory
.
A monoidal category \mathbf{C} which is symmetric and coclosed is called a symmetric coclosed monoidal category.
The corresponding GAP property is given by IsSymmetricCoclosedMonoidalCategory
.
A symmetric closed monoidal category \mathbf{C} satisfying
the natural morphism
\mathrm{\underline{Hom}}(a, a') \otimes \mathrm{\underline{Hom}}(b, b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b, a' \otimes b') is an isomorphism,
the natural morphism
a \rightarrow \mathrm{\underline{Hom}}(\mathrm{\underline{Hom}}(a, 1), 1) is an isomorphism is called a rigid symmetric closed monoidal category.
If no operations involving the closed structure are installed manually, the internal hom objects will be derived as \mathrm{\underline{Hom}}(a,b) \coloneqq a^\vee \otimes b and, in particular, \mathrm{\underline{Hom}}(a,1) \coloneqq a^\vee \otimes 1.
The corresponding GAP property is given by IsRigidSymmetricClosedMonoidalCategory
.
‣ IsomorphismFromTensorProductWithDualObjectToInternalHom ( 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{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).
‣ IsomorphismFromInternalHomToTensorProductWithDualObject ( 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{IsomorphismFromTensorProductWithDualObjectToInternalHom}, namely \mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes 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.
‣ TensorProductInternalHomCompatibilityMorphismInverse ( list ) | ( 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 argument is a list of 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 ( s, list, r ) | ( 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 a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') and r = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(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').
‣ 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}.
‣ TraceMap ( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(1,1).
The argument is an endomorphism \alpha: a \rightarrow a. The output is the trace morphism \mathrm{trace}_{\alpha}: 1 \rightarrow 1.
‣ 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.
‣ 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.
A symmetric coclosed monoidal category \mathbf{C} satisfying
the natural morphism
\mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a, b) \otimes \mathrm{\underline{coHom}}(a', b') is an isomorphism,
the natural morphism
\mathrm{\underline{coHom}}(1, \mathrm{\underline{coHom}}(1, a)) \rightarrow a is an isomorphism is called a rigid symmetric coclosed monoidal category.
If no operations involving the coclosed structure are installed manually, the internal cohom objects will be derived as \mathrm{\underline{coHom}}(a,b) \coloneqq a \otimes b_\vee and, in particular, \mathrm{\underline{coHom}}(1,a) \coloneqq 1 \otimes a_\vee.
The corresponding GAP property is given by IsRigidSymmetricCoclosedMonoidalCategory
.
‣ IsomorphismFromInternalCoHomToTensorProductWithCoDualObject ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).
The arguments are two objects a,b. The output is the natural morphism \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.
‣ IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).
The arguments are two objects a,b. The output is the inverse of \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}, namely \mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).
‣ MorphismFromTensorProductToInternalCoHom ( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a) ).
The arguments are two objects a,b. The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).
‣ MorphismFromTensorProductToInternalCoHomWithGivenObjects ( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).
The arguments are an object s_{\vee} = a \otimes b, two objects a,b, and an object r = \mathrm{\underline{coHom}}(b,a). The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).
‣ InternalCoHomTensorProductCompatibilityMorphismInverse ( list ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).
The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').
‣ InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects ( s, list, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).
The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') and r = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').
‣ CoclosedCoevaluationForCoDual ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).
The argument is an object a. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.
‣ CoclosedCoevaluationForCoDualWithGivenTensorProduct ( s, a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).
The arguments are an object s = a \otimes a_{\vee}, an object a, and an object r = 1. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.
‣ CoTraceMap ( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(1,1).
The argument is an endomorphism \alpha: a \rightarrow a. The output is the cotrace morphism \mathrm{cotrace}_{\alpha}: 1 \rightarrow 1.
‣ CoRankMorphism ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(1,1).
The argument is an object a. The output is the corank morphism \mathrm{corank}_a: 1 \rightarrow 1.
‣ MorphismToCoBidual ( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a, (a_{\vee})_{\vee}).
The argument is an object a. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.
‣ MorphismToCoBidualWithGivenCoBidual ( a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a,(a_{\vee})_{\vee}).
The argument is an object a, and an object r = (a_{\vee})_{\vee}. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.
‣ InternalHom ( a, b ) | ( operation ) |
Returns: a cell
This is a convenience method. The arguments are two cells a,b. The output is the internal hom cell. If a,b are two CAP objects the output is the internal Hom object \mathrm{\underline{Hom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal hom on morphisms, where any object is replaced by its identity morphism.
‣ InternalCoHom ( a, b ) | ( operation ) |
Returns: a cell
This is a convenience method. The arguments are two cells a,b. The output is the internal cohom cell. If a,b are two CAP objects the output is the internal cohom object \mathrm{\underline{coHom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal cohom on morphisms, where any object is replaced by its identity morphism.
‣ AddLeftDistributivityExpanding ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftDistributivityExpanding
. F: ( a, L ) \mapsto \mathtt{LeftDistributivityExpanding}(a, L).
‣ AddLeftDistributivityExpandingWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftDistributivityExpandingWithGivenObjects
. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r).
‣ AddLeftDistributivityFactoring ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftDistributivityFactoring
. F: ( a, L ) \mapsto \mathtt{LeftDistributivityFactoring}(a, L).
‣ AddLeftDistributivityFactoringWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftDistributivityFactoringWithGivenObjects
. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r).
‣ AddRightDistributivityExpanding ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightDistributivityExpanding
. F: ( L, a ) \mapsto \mathtt{RightDistributivityExpanding}(L, a).
‣ AddRightDistributivityExpandingWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightDistributivityExpandingWithGivenObjects
. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r).
‣ AddRightDistributivityFactoring ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightDistributivityFactoring
. F: ( L, a ) \mapsto \mathtt{RightDistributivityFactoring}(L, a).
‣ AddRightDistributivityFactoringWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightDistributivityFactoringWithGivenObjects
. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r).
‣ AddBraiding ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation Braiding
. F: ( a, b ) \mapsto \mathtt{Braiding}(a, b).
‣ AddBraidingInverse ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation BraidingInverse
. F: ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b).
‣ AddBraidingInverseWithGivenTensorProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation BraidingInverseWithGivenTensorProducts
. F: ( s, a, b, r ) \mapsto \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r).
‣ AddBraidingWithGivenTensorProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation BraidingWithGivenTensorProducts
. F: ( s, a, b, r ) \mapsto \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r).
‣ AddCoevaluationMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoevaluationMorphism
. F: ( a, b ) \mapsto \mathtt{CoevaluationMorphism}(a, b).
‣ AddCoevaluationMorphismWithGivenRange ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoevaluationMorphismWithGivenRange
. F: ( a, b, r ) \mapsto \mathtt{CoevaluationMorphismWithGivenRange}(a, b, r).
‣ AddDualOnMorphisms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DualOnMorphisms
. F: ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha).
‣ AddDualOnMorphismsWithGivenDuals ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DualOnMorphismsWithGivenDuals
. F: ( s, alpha, r ) \mapsto \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r).
‣ AddDualOnObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DualOnObjects
. F: ( a ) \mapsto \mathtt{DualOnObjects}(a).
‣ AddEvaluationForDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EvaluationForDual
. F: ( a ) \mapsto \mathtt{EvaluationForDual}(a).
‣ AddEvaluationForDualWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EvaluationForDualWithGivenTensorProduct
. F: ( s, a, r ) \mapsto \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r).
‣ AddEvaluationMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EvaluationMorphism
. F: ( a, b ) \mapsto \mathtt{EvaluationMorphism}(a, b).
‣ AddEvaluationMorphismWithGivenSource ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EvaluationMorphismWithGivenSource
. F: ( a, b, s ) \mapsto \mathtt{EvaluationMorphismWithGivenSource}(a, b, s).
‣ AddInternalHomOnMorphisms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalHomOnMorphisms
. F: ( alpha, beta ) \mapsto \mathtt{InternalHomOnMorphisms}(alpha, beta).
‣ AddInternalHomOnMorphismsWithGivenInternalHoms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalHomOnMorphismsWithGivenInternalHoms
. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta, r).
‣ AddInternalHomOnObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalHomOnObjects
. F: ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a, b).
‣ AddInternalHomToTensorProductAdjunctionMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalHomToTensorProductAdjunctionMap
. F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMap}(b, c, g).
‣ AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct
. F: ( b, c, g, t ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct}(b, c, g, t).
‣ AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromDualObjectToInternalHomIntoTensorUnit
. F: ( a ) \mapsto \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a).
‣ AddIsomorphismFromInternalHomIntoTensorUnitToDualObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalHomIntoTensorUnitToDualObject
. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a).
‣ AddIsomorphismFromInternalHomToObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalHomToObject
. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomToObject}(a).
‣ AddIsomorphismFromInternalHomToObjectWithGivenInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalHomToObjectWithGivenInternalHom
. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s).
‣ AddIsomorphismFromObjectToInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToInternalHom
. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalHom}(a).
‣ AddIsomorphismFromObjectToInternalHomWithGivenInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToInternalHomWithGivenInternalHom
. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r).
‣ AddLambdaElimination ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LambdaElimination
. F: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha).
‣ AddLambdaIntroduction ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LambdaIntroduction
. F: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha).
‣ AddMonoidalPostComposeMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPostComposeMorphism
. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostComposeMorphism}(a, b, c).
‣ AddMonoidalPostComposeMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPostComposeMorphismWithGivenObjects
. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddMonoidalPreComposeMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPreComposeMorphism
. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreComposeMorphism}(a, b, c).
‣ AddMonoidalPreComposeMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPreComposeMorphismWithGivenObjects
. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddMorphismFromTensorProductToInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromTensorProductToInternalHom
. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalHom}(a, b).
‣ AddMorphismFromTensorProductToInternalHomWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromTensorProductToInternalHomWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s, a, b, r).
‣ AddMorphismToBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToBidual
. F: ( a ) \mapsto \mathtt{MorphismToBidual}(a).
‣ AddMorphismToBidualWithGivenBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToBidualWithGivenBidual
. F: ( a, r ) \mapsto \mathtt{MorphismToBidualWithGivenBidual}(a, r).
‣ AddTensorProductDualityCompatibilityMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductDualityCompatibilityMorphism
. F: ( a, b ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphism}(a, b).
‣ AddTensorProductDualityCompatibilityMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductDualityCompatibilityMorphismWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).
‣ AddTensorProductInternalHomCompatibilityMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductInternalHomCompatibilityMorphism
. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphism}(list).
‣ AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductInternalHomCompatibilityMorphismWithGivenObjects
. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range).
‣ AddTensorProductToInternalHomAdjunctionMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductToInternalHomAdjunctionMap
. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMap}(a, b, f).
‣ AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductToInternalHomAdjunctionMapWithGivenInternalHom
. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMapWithGivenInternalHom}(a, b, f, i).
‣ AddUniversalPropertyOfDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalPropertyOfDual
. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfDual}(t, a, alpha).
‣ AddCoDualOnMorphisms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoDualOnMorphisms
. F: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha).
‣ AddCoDualOnMorphismsWithGivenCoDuals ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoDualOnMorphismsWithGivenCoDuals
. F: ( s, alpha, r ) \mapsto \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r).
‣ AddCoDualOnObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoDualOnObjects
. F: ( a ) \mapsto \mathtt{CoDualOnObjects}(a).
‣ AddCoDualityTensorProductCompatibilityMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoDualityTensorProductCompatibilityMorphism
. F: ( a, b ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b).
‣ AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoDualityTensorProductCompatibilityMorphismWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r).
‣ AddCoLambdaElimination ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoLambdaElimination
. F: ( a, b, alpha ) \mapsto \mathtt{CoLambdaElimination}(a, b, alpha).
‣ AddCoLambdaIntroduction ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoLambdaIntroduction
. F: ( alpha ) \mapsto \mathtt{CoLambdaIntroduction}(alpha).
‣ AddCoclosedCoevaluationMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedCoevaluationMorphism
. F: ( a, b ) \mapsto \mathtt{CoclosedCoevaluationMorphism}(a, b).
‣ AddCoclosedCoevaluationMorphismWithGivenSource ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedCoevaluationMorphismWithGivenSource
. F: ( a, b, s ) \mapsto \mathtt{CoclosedCoevaluationMorphismWithGivenSource}(a, b, s).
‣ AddCoclosedEvaluationForCoDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedEvaluationForCoDual
. F: ( a ) \mapsto \mathtt{CoclosedEvaluationForCoDual}(a).
‣ AddCoclosedEvaluationForCoDualWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedEvaluationForCoDualWithGivenTensorProduct
. F: ( s, a, r ) \mapsto \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r).
‣ AddCoclosedEvaluationMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedEvaluationMorphism
. F: ( a, b ) \mapsto \mathtt{CoclosedEvaluationMorphism}(a, b).
‣ AddCoclosedEvaluationMorphismWithGivenRange ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedEvaluationMorphismWithGivenRange
. F: ( a, b, r ) \mapsto \mathtt{CoclosedEvaluationMorphismWithGivenRange}(a, b, r).
‣ AddInternalCoHomOnMorphisms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomOnMorphisms
. F: ( alpha, beta ) \mapsto \mathtt{InternalCoHomOnMorphisms}(alpha, beta).
‣ AddInternalCoHomOnMorphismsWithGivenInternalCoHoms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomOnMorphismsWithGivenInternalCoHoms
. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s, alpha, beta, r).
‣ AddInternalCoHomOnObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomOnObjects
. F: ( a, b ) \mapsto \mathtt{InternalCoHomOnObjects}(a, b).
‣ AddInternalCoHomTensorProductCompatibilityMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphism
. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list).
‣ AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range).
‣ AddInternalCoHomToTensorProductAdjunctionMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomToTensorProductAdjunctionMap
. F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMap}(a, b, f).
‣ AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct
. F: ( a, b, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct}(a, b, f, t).
‣ AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
. F: ( a ) \mapsto \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a).
‣ AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a).
‣ AddIsomorphismFromInternalCoHomToObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalCoHomToObject
. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObject}(a).
‣ AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s).
‣ AddIsomorphismFromObjectToInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToInternalCoHom
. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHom}(a).
‣ AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r).
‣ AddMonoidalPostCoComposeMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPostCoComposeMorphism
. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostCoComposeMorphism}(a, b, c).
‣ AddMonoidalPostCoComposeMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPostCoComposeMorphismWithGivenObjects
. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddMonoidalPreCoComposeMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPreCoComposeMorphism
. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreCoComposeMorphism}(a, b, c).
‣ AddMonoidalPreCoComposeMorphismWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonoidalPreCoComposeMorphismWithGivenObjects
. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddMorphismFromCoBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromCoBidual
. F: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a).
‣ AddMorphismFromCoBidualWithGivenCoBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromCoBidualWithGivenCoBidual
. F: ( a, s ) \mapsto \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s).
‣ AddMorphismFromInternalCoHomToTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromInternalCoHomToTensorProduct
. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b).
‣ AddMorphismFromInternalCoHomToTensorProductWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromInternalCoHomToTensorProductWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r).
‣ AddTensorProductToInternalCoHomAdjunctionMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductToInternalCoHomAdjunctionMap
. F: ( c, b, g ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMap}(c, b, g).
‣ AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom
. F: ( c, b, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom}(c, b, g, i).
‣ AddUniversalPropertyOfCoDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalPropertyOfCoDual
. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCoDual}(t, a, alpha).
‣ AddAssociatorLeftToRight ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation AssociatorLeftToRight
. F: ( a, b, c ) \mapsto \mathtt{AssociatorLeftToRight}(a, b, c).
‣ AddAssociatorLeftToRightWithGivenTensorProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation AssociatorLeftToRightWithGivenTensorProducts
. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r).
‣ AddAssociatorRightToLeft ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation AssociatorRightToLeft
. F: ( a, b, c ) \mapsto \mathtt{AssociatorRightToLeft}(a, b, c).
‣ AddAssociatorRightToLeftWithGivenTensorProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation AssociatorRightToLeftWithGivenTensorProducts
. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r).
‣ AddLeftUnitor ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftUnitor
. F: ( a ) \mapsto \mathtt{LeftUnitor}(a).
‣ AddLeftUnitorInverse ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftUnitorInverse
. F: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a).
‣ AddLeftUnitorInverseWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftUnitorInverseWithGivenTensorProduct
. F: ( a, r ) \mapsto \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r).
‣ AddLeftUnitorWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftUnitorWithGivenTensorProduct
. F: ( a, s ) \mapsto \mathtt{LeftUnitorWithGivenTensorProduct}(a, s).
‣ AddRightUnitor ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightUnitor
. F: ( a ) \mapsto \mathtt{RightUnitor}(a).
‣ AddRightUnitorInverse ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightUnitorInverse
. F: ( a ) \mapsto \mathtt{RightUnitorInverse}(a).
‣ AddRightUnitorInverseWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightUnitorInverseWithGivenTensorProduct
. F: ( a, r ) \mapsto \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r).
‣ AddRightUnitorWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightUnitorWithGivenTensorProduct
. F: ( a, s ) \mapsto \mathtt{RightUnitorWithGivenTensorProduct}(a, s).
‣ AddTensorProductOnMorphisms ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductOnMorphisms
. F: ( alpha, beta ) \mapsto \mathtt{TensorProductOnMorphisms}(alpha, beta).
‣ AddTensorProductOnMorphismsWithGivenTensorProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductOnMorphismsWithGivenTensorProducts
. F: ( s, alpha, beta, r ) \mapsto \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s, alpha, beta, r).
‣ AddCoevaluationForDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoevaluationForDual
. F: ( a ) \mapsto \mathtt{CoevaluationForDual}(a).
‣ AddCoevaluationForDualWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoevaluationForDualWithGivenTensorProduct
. F: ( s, a, r ) \mapsto \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r).
‣ AddIsomorphismFromInternalHomToTensorProductWithDualObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalHomToTensorProductWithDualObject
. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b).
‣ AddIsomorphismFromTensorProductWithDualObjectToInternalHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromTensorProductWithDualObjectToInternalHom
. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b).
‣ AddMorphismFromBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromBidual
. F: ( a ) \mapsto \mathtt{MorphismFromBidual}(a).
‣ AddMorphismFromBidualWithGivenBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromBidualWithGivenBidual
. F: ( a, s ) \mapsto \mathtt{MorphismFromBidualWithGivenBidual}(a, s).
‣ AddMorphismFromInternalHomToTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromInternalHomToTensorProduct
. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalHomToTensorProduct}(a, b).
‣ AddMorphismFromInternalHomToTensorProductWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromInternalHomToTensorProductWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s, a, b, r).
‣ AddRankMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RankMorphism
. F: ( a ) \mapsto \mathtt{RankMorphism}(a).
‣ AddTensorProductInternalHomCompatibilityMorphismInverse ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductInternalHomCompatibilityMorphismInverse
. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list).
‣ AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source, list, range).
‣ AddTraceMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation TraceMap
. F: ( alpha ) \mapsto \mathtt{TraceMap}(alpha).
‣ AddCoRankMorphism ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoRankMorphism
. F: ( a ) \mapsto \mathtt{CoRankMorphism}(a).
‣ AddCoTraceMap ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoTraceMap
. F: ( alpha ) \mapsto \mathtt{CoTraceMap}(alpha).
‣ AddCoclosedCoevaluationForCoDual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedCoevaluationForCoDual
. F: ( a ) \mapsto \mathtt{CoclosedCoevaluationForCoDual}(a).
‣ AddCoclosedCoevaluationForCoDualWithGivenTensorProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CoclosedCoevaluationForCoDualWithGivenTensorProduct
. F: ( s, a, r ) \mapsto \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r).
‣ AddInternalCoHomTensorProductCompatibilityMorphismInverse ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismInverse
. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list).
‣ AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source, list, range).
‣ AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b).
‣ AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b).
‣ AddMorphismFromTensorProductToInternalCoHom ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromTensorProductToInternalCoHom
. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b).
‣ AddMorphismFromTensorProductToInternalCoHomWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromTensorProductToInternalCoHomWithGivenObjects
. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s, a, b, r).
‣ AddMorphismToCoBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToCoBidual
. F: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a).
‣ AddMorphismToCoBidualWithGivenCoBidual ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToCoBidualWithGivenCoBidual
. F: ( a, r ) \mapsto \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r).
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