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}_\ell}(b,-)) is called a left closed monoidal category.
If no operations involving left duals are installed manually, the left dual objects will be derived as a^\vee \coloneqq \mathrm{\underline{Hom}_\ell}(a,1).
The corresponding GAP property is called IsLeftClosedMonoidalCategory.
‣ LeftInternalHomOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects a,b. The output is the internal hom object \mathrm{\underline{Hom}_\ell}(a,b).
‣ LeftInternalHomOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a',b), \mathrm{\underline{Hom}_\ell}(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}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b').
‣ LeftInternalHomOnMorphismsWithGivenLeftInternalHoms( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r )
The arguments are an object s = \mathrm{\underline{Hom}_\ell}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{Hom}_\ell}(a,b'). The output is the internal hom morphism \mathrm{\underline{Hom}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b').
‣ LeftClosedMonoidalEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(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}_\ell}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ LeftClosedMonoidalEvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes a. The output is the evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ LeftClosedMonoidalCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}_\ell}(a, b \otimes a) ).
The arguments are two objects a,b. The output is the coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.
‣ LeftClosedMonoidalCoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}_\ell}(a, b \otimes a). The output is the coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.
‣ TensorProductToLeftInternalHomAdjunctMorphism( a, b, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}_\ell}(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}_\ell}(b,c) corresponding to f under the tensor hom adjunction.
‣ TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom( a, b, f, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, i ).
The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}_\ell}(b,c). The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c) corresponding to f under the tensor hom adjunction.
‣ LeftInternalHomToTensorProductAdjunctMorphism( 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}_\ell}(b,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.
‣ LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct( b, c, g, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(t, c).
The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(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.
‣ LeftClosedMonoidalPreComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c), \mathrm{\underline{Hom}_\ell}(a,c) ).
The arguments are three objects a,b,c. The output is the precomposition morphism \mathrm{LeftClosedMonoidalPreComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).
‣ LeftClosedMonoidalPreComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c), three objects a,b,c, and an object r = \mathrm{\underline{Hom}_\ell}(a,c). The output is the precomposition morphism \mathrm{LeftClosedMonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).
‣ LeftClosedMonoidalPostComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b), \mathrm{\underline{Hom}_\ell}(a,c) ).
The arguments are three objects a,b,c. The output is the postcomposition morphism \mathrm{LeftClosedMonoidalPostComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).
‣ LeftClosedMonoidalPostComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b), three objects a,b,c, and an object r = \mathrm{\underline{Hom}_\ell}(a,c). The output is the postcomposition morphism \mathrm{LeftClosedMonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).
‣ LeftDualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object a. The output is its dual object a^{\vee}.
‣ LeftDualOnMorphisms( 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}.
‣ LeftDualOnMorphismsWithGivenLeftDuals( s, alpha, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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}.
‣ LeftClosedMonoidalEvaluationForLeftDual( 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.
‣ LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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.
‣ MorphismToLeftBidual( 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}.
‣ MorphismToLeftBidualWithGivenLeftBidual( a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, r).
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}.
‣ TensorProductLeftInternalHomCompatibilityMorphism( list ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b'), \mathrm{\underline{Hom}_\ell}(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{TensorProductLeftInternalHomCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b').
‣ TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') and r = \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b'). The output is the natural morphism \mathrm{TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b').
‣ TensorProductLeftDualityCompatibilityMorphism( 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{TensorProductLeftDualityCompatibilityMorphism}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.
‣ TensorProductLeftDualityCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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{TensorProductLeftDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.
‣ MorphismFromTensorProductToLeftInternalHom( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}_\ell}(a,b) ).
The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromTensorProductToLeftInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}_\ell}(a,b).
‣ MorphismFromTensorProductToLeftInternalHomWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = a^{\vee} \otimes b, two objects a,b, and an object r = \mathrm{\underline{Hom}_\ell}(a,b). The output is the natural morphism \mathrm{MorphismFromTensorProductToLeftInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}_\ell}(a,b).
‣ IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}_\ell}(a,1)).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}_\ell}(a,1).
‣ IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}_\ell}(a,1), a^{\vee}).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject}_{a}: \mathrm{\underline{Hom}_\ell}(a,1) \rightarrow a^{\vee}.
‣ UniversalPropertyOfLeftDual( 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}.
‣ LeftClosedMonoidalLambdaIntroduction( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}( 1, \mathrm{\underline{Hom}_\ell}(a,b) ).
The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism 1 \rightarrow \mathrm{\underline{Hom}_\ell}(a,b) under the tensor hom adjunction.
‣ LeftClosedMonoidalLambdaElimination( 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}_\ell}(a,b). The output is a morphism a \rightarrow b corresponding to \alpha under the tensor hom adjunction.
‣ IsomorphismFromObjectToLeftInternalHom( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{Hom}_\ell}(1,a)).
The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}_\ell}(1,a).
‣ IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom( a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, r).
The argument is an object a, and an object r = \mathrm{\underline{Hom}_\ell}(1,a). The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}_\ell}(1,a).
‣ IsomorphismFromLeftInternalHomToObject( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}_\ell}(1,a),a).
The argument is an object a. The output is the natural isomorphism \mathrm{\underline{Hom}_\ell}(1,a) \rightarrow a.
‣ IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom( a, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s,a).
The argument is an object a, and an object s = \mathrm{\underline{Hom}_\ell}(1,a). The output is the natural isomorphism \mathrm{\underline{Hom}_\ell}(1,a) \rightarrow a.
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}_\ell}(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}_\ell}(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}( s, r )
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').
‣ ClosedMonoidalRightEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a \otimes \mathrm{\underline{Hom}}(a,b), b ).
The arguments are two objects a, b. The output is the right evaluation morphism \mathrm{ev}_{a,b}:a \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ ClosedMonoidalRightEvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = a \otimes \mathrm{\underline{Hom}}(a,b). The output is the right evaluation morphism \mathrm{ev}_{a,b}: a \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ ClosedMonoidalRightCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a, a \otimes b) ).
The arguments are two objects a,b. The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, a \otimes b), i.e., the unit of the tensor hom adjunction.
‣ ClosedMonoidalRightCoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}}(a, a \otimes b). The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, a \otimes b), i.e., the unit of the tensor hom adjunction.
‣ TensorProductToInternalHomRightAdjunctMorphism( a, b, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a,c) ).
The arguments are two objects a,b and a morphism f: a \otimes b \rightarrow c. The output is a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c) corresponding to f under the tensor hom adjunction.
‣ TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom( a, b, f, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, i ).
The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}}(a,c). The output is a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c) corresponding to f under the tensor hom adjunction.
‣ InternalHomToTensorProductRightAdjunctMorphism( a, c, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a \otimes b, c).
The arguments are two objects a,c and a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.
‣ InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct( a, c, g, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, c).
The arguments are two objects a,c, a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c) and an object s = a \otimes b. The output is a morphism f: s \rightarrow c corresponding to g under the tensor hom adjunction.
‣ ClosedMonoidalLeftEvaluationMorphism( 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 left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ ClosedMonoidalLeftEvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{Hom}}(a,b) \otimes a. The output is the left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.
‣ ClosedMonoidalLeftCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a, b \otimes a) ).
The arguments are two objects a,b. The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.
‣ ClosedMonoidalLeftCoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}}(a, b \otimes a). The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.
‣ TensorProductToInternalHomLeftAdjunctMorphism( 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.
‣ TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom( a, b, f, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, i ).
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.
‣ InternalHomToTensorProductLeftAdjunctMorphism( 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.
‣ InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( b, c, g, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, c).
The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) and an object s = a \otimes b. The output is a morphism f: s \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{MonoidalPreComposeMorphism}_{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}( s, r ).
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{MonoidalPostComposeMorphism}_{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}( s, r ).
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}( s, r ).
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}( s, r ).
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, r).
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{TensorProductInternalHomCompatibilityMorphism}_{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}( s, r ).
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{TensorProductDualityCompatibilityMorphism}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.
‣ TensorProductDualityCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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{MorphismFromTensorProductToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).
‣ MorphismFromTensorProductToInternalHomWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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, r).
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}(s,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 left coclosed monoidal category.
If no operations involving left coduals are installed manually, the left codual objects will be derived as a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a).
The corresponding GAP property is called IsLeftCoclosedMonoidalCategory.
‣ LeftInternalCoHomOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}_\ell}(a,b).
‣ LeftInternalCoHomOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b'), \mathrm{\underline{coHom}_\ell}(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}_\ell}(\alpha,\beta): \mathrm{\underline{coHom}_\ell}(a,b') \rightarrow \mathrm{\underline{coHom}_\ell}(a',b).
‣ LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r )
The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,b'), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{coHom}_\ell}(a',b). The output is the internal cohom morphism \mathrm{\underline{coHom}_\ell}(\alpha,\beta): \mathrm{\underline{coHom}_\ell}(a,b') \rightarrow \mathrm{\underline{coHom}_\ell}(a',b).
‣ LeftCoclosedMonoidalEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{coHom}_\ell}(b,a) \otimes a ).
The arguments are two objects a, b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}_\ell}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.
‣ LeftCoclosedMonoidalEvaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}_\ell}(b,a) \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}_\ell}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.
‣ LeftCoclosedMonoidalCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(b \otimes a, a), b ).
The arguments are two objects a,b. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}_\ell}(b \otimes a, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.
‣ LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}_\ell}(b \otimes a, a). The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}_\ell}(b \otimes a, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.
‣ TensorProductToLeftInternalCoHomAdjunctMorphism( b, c, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), b ).
The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.
‣ TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom( b, c, g, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( i, b ).
The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c and an object i = \mathrm{\underline{coHom}_\ell}(a,c). The output is a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.
‣ LeftInternalCoHomToTensorProductAdjunctMorphism( a, c, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, b \otimes c).
The arguments are two objects a,c and a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.
‣ LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct( a, c, f, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, t ).
The arguments are two objects a,c, a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b and an object t = b \otimes c. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.
‣ LeftCoclosedMonoidalPreCoComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b) ).
The arguments are three objects a,b,c. The output is the precocomposition morphism \mathrm{LeftCoclosedMonoidalPreCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b).
‣ LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c). The output is the precocomposition morphism \mathrm{LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b).
‣ LeftCoclosedMonoidalPostCoComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c) ).
The arguments are three objects a,b,c. The output is the postcocomposition morphism \mathrm{LeftCoclosedMonoidalPostCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c).
‣ LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b). The output is the postcocomposition morphism \mathrm{LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c).
‣ LeftCoDualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object a. The output is its codual object a_{\vee}.
‣ LeftCoDualOnMorphisms( 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}.
‣ LeftCoDualOnMorphismsWithGivenLeftCoDuals( s, alpha, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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}.
‣ LeftCoclosedMonoidalEvaluationForLeftCoDual( 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.
‣ LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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.
‣ MorphismFromLeftCoBidual( 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.
‣ MorphismFromLeftCoBidualWithGivenLeftCoBidual( a, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, 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.
‣ LeftInternalCoHomTensorProductCompatibilityMorphism( list ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b')).
The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{LeftInternalCoHomTensorProductCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b').
‣ LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') and r = \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b'). The output is the natural morphism \mathrm{LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b').
‣ LeftCoDualityTensorProductCompatibilityMorphism( 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{LeftCoDualityTensorProductCompatibilityMorphism}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.
‣ LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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{LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.
‣ MorphismFromLeftInternalCoHomToTensorProduct( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b), b_{\vee} \otimes a ).
The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromLeftInternalCoHomToTensorProduct}_{a,b}: \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow b_{\vee} \otimes a.
‣ MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,b), two objects a,b, and an object r = b_{\vee} \otimes a. The output is the natural morphism \mathrm{MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow a \otimes b_{\vee}.
‣ IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}_\ell}(1,a)).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}_\ell}(1,a).
‣ IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}_\ell}(1,a), a_{\vee}).
The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject}_{a}: \mathrm{\underline{coHom}_\ell}(1,a) \rightarrow a_{\vee}.
‣ UniversalPropertyOfLeftCoDual( 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}.
‣ LeftCoclosedMonoidalLambdaIntroduction( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b), 1 ).
The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow 1 under the cohom tensor adjunction.
‣ LeftCoclosedMonoidalLambdaElimination( 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}_\ell}(a,b) \rightarrow 1. The output is a morphism a \rightarrow b corresponding to \alpha under the cohom tensor adjunction.
‣ IsomorphismFromObjectToLeftInternalCoHom( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}_\ell}(a,1)).
The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}_\ell}(a,1).
‣ IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom( a, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, r).
The argument is an object a, and an object r = \mathrm{\underline{coHom}_\ell}(a,1). The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}_\ell}(a,1).
‣ IsomorphismFromLeftInternalCoHomToObject( a ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}_\ell}(a,1), a).
The argument is an object a. The output is the natural isomorphism \mathrm{\underline{coHom}_\ell}(a,1) \rightarrow a.
‣ IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom( a, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, a).
The argument is an object a, and an object s = \mathrm{\underline{coHom}_\ell}(a,1). The output is the natural isomorphism \mathrm{\underline{coHom}_\ell}(a,1) \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}( s, r )
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).
‣ CoclosedMonoidalRightEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, a \otimes \mathrm{\underline{coHom}}(b,a) ).
The arguments are two objects a, b. The output is the coclosed right evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow a \otimes \mathrm{\underline{coHom}}(b,a), i.e., the unit of the cohom tensor adjunction.
‣ CoclosedMonoidalRightEvaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = a \otimes \mathrm{\underline{coHom}}(b,a). The output is the coclosed right evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow a \otimes \mathrm{\underline{coHom}}(b,a), i.e., the unit of the cohom tensor adjunction.
‣ CoclosedMonoidalRightCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, a), b ).
The arguments are two objects a,b. The output is the coclosed right coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.
‣ CoclosedMonoidalRightCoevaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}}(a \otimes b, a). The output is the coclosed right coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.
‣ TensorProductToInternalCoHomRightAdjunctMorphism( b, c, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).
The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.
‣ TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom( b, c, g, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( i, c ).
The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c 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.
‣ InternalCoHomToTensorProductRightAdjunctMorphism( a, b, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, b \otimes c).
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 b \otimes c corresponding to f under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct( a, b, f, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, t ).
The arguments are two objects a,b, a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c and an object t = b \otimes c. The output is a morphism g: a \rightarrow t corresponding to f under the cohom tensor adjunction.
‣ CoclosedMonoidalLeftEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{coHom}}(b,a) \otimes a ).
The arguments are two objects a, b. The output is the coclosed left evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.
‣ CoclosedMonoidalLeftEvaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( b, r ).
The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}}(b,a) \otimes a. The output is the coclosed left evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.
‣ CoclosedMonoidalLeftCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(b \otimes a, a), b ).
The arguments are two objects a,b. The output is the coclosed left coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(b \otimes a, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.
‣ CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, b ).
The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}}(b \otimes a, a). The output is the coclosed left coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(b \otimes a, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.
‣ TensorProductToInternalCoHomLeftAdjunctMorphism( b, c, g ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), b ).
The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.
‣ TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom( b, c, g, i ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( i, b ).
The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c and an object i = \mathrm{\underline{coHom}}(a,c). The output is a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductLeftAdjunctMorphism( a, c, f ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(a, b \otimes c).
The arguments are two objects a,c and a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( a, c, f, t ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( a, t ).
The arguments are two objects a,c, a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b and an object t = b \otimes c. The output is a morphism g: a \rightarrow t 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{MonoidalPreCoComposeMorphism}_{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}( s, r ).
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{MonoidalPostCoComposeMorphism}_{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}( s, r ).
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}( s, r ).
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}( s, r ).
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}(s, 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{InternalCoHomTensorProductCompatibilityMorphism}_{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}( s, r ).
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{CoDualityTensorProductCompatibilityMorphism}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.
‣ CoDualityTensorProductCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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{MorphismFromInternalCoHomToTensorProduct}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.
‣ MorphismFromInternalCoHomToTensorProductWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( s, r ).
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, r).
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}(s, 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}_\ell}(a, a') \otimes \mathrm{\underline{Hom}_\ell}(b, b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b, a' \otimes b') is an isomorphism,
the natural morphism
a \rightarrow \mathrm{\underline{Hom}_\ell}(\mathrm{\underline{Hom}_\ell}(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}_\ell}(a,b) \coloneqq a^\vee \otimes b and, in particular, \mathrm{\underline{Hom}_\ell}(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.
‣ LeftInternalHom( 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}_\ell}(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.
‣ LeftInternalCoHom( 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}_\ell}(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).
‣ AddClosedMonoidalLeftCoevaluationMorphism( 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 ClosedMonoidalLeftCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalLeftCoevaluationMorphism}(a, b).
‣ AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange( 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 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{ClosedMonoidalLeftCoevaluationMorphismWithGivenRange}(a, b, r).
‣ AddClosedMonoidalLeftEvaluationMorphism( 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 ClosedMonoidalLeftEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalLeftEvaluationMorphism}(a, b).
‣ AddClosedMonoidalLeftEvaluationMorphismWithGivenSource( 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 ClosedMonoidalLeftEvaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{ClosedMonoidalLeftEvaluationMorphismWithGivenSource}(a, b, s).
‣ AddClosedMonoidalRightCoevaluationMorphism( 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 ClosedMonoidalRightCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalRightCoevaluationMorphism}(a, b).
‣ AddClosedMonoidalRightCoevaluationMorphismWithGivenRange( 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 ClosedMonoidalRightCoevaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{ClosedMonoidalRightCoevaluationMorphismWithGivenRange}(a, b, r).
‣ AddClosedMonoidalRightEvaluationMorphism( 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 ClosedMonoidalRightEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalRightEvaluationMorphism}(a, b).
‣ AddClosedMonoidalRightEvaluationMorphismWithGivenSource( 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 ClosedMonoidalRightEvaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{ClosedMonoidalRightEvaluationMorphismWithGivenSource}(a, b, s).
‣ 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).
‣ 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).
‣ AddInternalHomToTensorProductLeftAdjunctMorphism( 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 InternalHomToTensorProductLeftAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctMorphism}(b, c, g).
‣ AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( 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 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct. F: ( b, c, g, s ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct}(b, c, g, s).
‣ AddInternalHomToTensorProductRightAdjunctMorphism( 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 InternalHomToTensorProductRightAdjunctMorphism. F: ( a, c, g ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctMorphism}(a, c, g).
‣ AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct( 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 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct. F: ( a, c, g, s ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct}(a, c, g, s).
‣ 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).
‣ AddTensorProductToInternalHomLeftAdjunctMorphism( 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 TensorProductToInternalHomLeftAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctMorphism}(a, b, f).
‣ AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom( 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 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom}(a, b, f, i).
‣ AddTensorProductToInternalHomRightAdjunctMorphism( 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 TensorProductToInternalHomRightAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctMorphism}(a, b, f).
‣ AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom( 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 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom}(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).
‣ 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).
‣ AddCoclosedMonoidalLeftCoevaluationMorphism( 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 CoclosedMonoidalLeftCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalLeftCoevaluationMorphism}(a, b).
‣ AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource( 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 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource}(a, b, s).
‣ AddCoclosedMonoidalLeftEvaluationMorphism( 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 CoclosedMonoidalLeftEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalLeftEvaluationMorphism}(a, b).
‣ AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange( 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 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{CoclosedMonoidalLeftEvaluationMorphismWithGivenRange}(a, b, r).
‣ AddCoclosedMonoidalRightCoevaluationMorphism( 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 CoclosedMonoidalRightCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalRightCoevaluationMorphism}(a, b).
‣ AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource( 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 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{CoclosedMonoidalRightCoevaluationMorphismWithGivenSource}(a, b, s).
‣ AddCoclosedMonoidalRightEvaluationMorphism( 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 CoclosedMonoidalRightEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalRightEvaluationMorphism}(a, b).
‣ AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange( 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 CoclosedMonoidalRightEvaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{CoclosedMonoidalRightEvaluationMorphismWithGivenRange}(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).
‣ AddInternalCoHomToTensorProductLeftAdjunctMorphism( 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 InternalCoHomToTensorProductLeftAdjunctMorphism. F: ( a, c, f ) \mapsto \mathtt{InternalCoHomToTensorProductLeftAdjunctMorphism}(a, c, f).
‣ AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( 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 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct. F: ( a, c, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct}(a, c, f, t).
‣ AddInternalCoHomToTensorProductRightAdjunctMorphism( 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 InternalCoHomToTensorProductRightAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductRightAdjunctMorphism}(a, b, f).
‣ AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct( 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 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct. F: ( a, b, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct}(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).
‣ AddTensorProductToInternalCoHomLeftAdjunctMorphism( 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 TensorProductToInternalCoHomLeftAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{TensorProductToInternalCoHomLeftAdjunctMorphism}(b, c, g).
‣ AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom( 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 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom}(b, c, g, i).
‣ AddTensorProductToInternalCoHomRightAdjunctMorphism( 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 TensorProductToInternalCoHomRightAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{TensorProductToInternalCoHomRightAdjunctMorphism}(b, c, g).
‣ AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom( 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 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom}(b, c, 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).
‣ AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit( 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 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit}(a).
‣ AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject( 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 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject}(a).
‣ AddIsomorphismFromLeftInternalHomToObject( 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 IsomorphismFromLeftInternalHomToObject. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalHomToObject}(a).
‣ AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom( 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 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom. F: ( a, s ) \mapsto \mathtt{IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom}(a, s).
‣ AddIsomorphismFromObjectToLeftInternalHom( 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 IsomorphismFromObjectToLeftInternalHom. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalHom}(a).
‣ AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom( 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 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom}(a, r).
‣ AddLeftClosedMonoidalCoevaluationMorphism( 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 LeftClosedMonoidalCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{LeftClosedMonoidalCoevaluationMorphism}(a, b).
‣ AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange( 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 LeftClosedMonoidalCoevaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{LeftClosedMonoidalCoevaluationMorphismWithGivenRange}(a, b, r).
‣ AddLeftClosedMonoidalEvaluationForLeftDual( 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 LeftClosedMonoidalEvaluationForLeftDual. F: ( a ) \mapsto \mathtt{LeftClosedMonoidalEvaluationForLeftDual}(a).
‣ AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct( 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 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct. F: ( s, a, r ) \mapsto \mathtt{LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct}(s, a, r).
‣ AddLeftClosedMonoidalEvaluationMorphism( 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 LeftClosedMonoidalEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{LeftClosedMonoidalEvaluationMorphism}(a, b).
‣ AddLeftClosedMonoidalEvaluationMorphismWithGivenSource( 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 LeftClosedMonoidalEvaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{LeftClosedMonoidalEvaluationMorphismWithGivenSource}(a, b, s).
‣ AddLeftClosedMonoidalLambdaElimination( 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 LeftClosedMonoidalLambdaElimination. F: ( a, b, alpha ) \mapsto \mathtt{LeftClosedMonoidalLambdaElimination}(a, b, alpha).
‣ AddLeftClosedMonoidalLambdaIntroduction( 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 LeftClosedMonoidalLambdaIntroduction. F: ( alpha ) \mapsto \mathtt{LeftClosedMonoidalLambdaIntroduction}(alpha).
‣ AddLeftClosedMonoidalPostComposeMorphism( 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 LeftClosedMonoidalPostComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{LeftClosedMonoidalPostComposeMorphism}(a, b, c).
‣ AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects( 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 LeftClosedMonoidalPostComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftClosedMonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddLeftClosedMonoidalPreComposeMorphism( 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 LeftClosedMonoidalPreComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{LeftClosedMonoidalPreComposeMorphism}(a, b, c).
‣ AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects( 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 LeftClosedMonoidalPreComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftClosedMonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddLeftDualOnMorphisms( 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 LeftDualOnMorphisms. F: ( alpha ) \mapsto \mathtt{LeftDualOnMorphisms}(alpha).
‣ AddLeftDualOnMorphismsWithGivenLeftDuals( 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 LeftDualOnMorphismsWithGivenLeftDuals. F: ( s, alpha, r ) \mapsto \mathtt{LeftDualOnMorphismsWithGivenLeftDuals}(s, alpha, r).
‣ AddLeftDualOnObjects( 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 LeftDualOnObjects. F: ( a ) \mapsto \mathtt{LeftDualOnObjects}(a).
‣ AddLeftInternalHomOnMorphisms( 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 LeftInternalHomOnMorphisms. F: ( alpha, beta ) \mapsto \mathtt{LeftInternalHomOnMorphisms}(alpha, beta).
‣ AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms( 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 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms. F: ( s, alpha, beta, r ) \mapsto \mathtt{LeftInternalHomOnMorphismsWithGivenLeftInternalHoms}(s, alpha, beta, r).
‣ AddLeftInternalHomOnObjects( 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 LeftInternalHomOnObjects. F: ( a, b ) \mapsto \mathtt{LeftInternalHomOnObjects}(a, b).
‣ AddLeftInternalHomToTensorProductAdjunctMorphism( 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 LeftInternalHomToTensorProductAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{LeftInternalHomToTensorProductAdjunctMorphism}(b, c, g).
‣ AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct( 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 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct. F: ( b, c, g, t ) \mapsto \mathtt{LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct}(b, c, g, t).
‣ AddMorphismFromTensorProductToLeftInternalHom( 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 MorphismFromTensorProductToLeftInternalHom. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToLeftInternalHom}(a, b).
‣ AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects( 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 MorphismFromTensorProductToLeftInternalHomWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToLeftInternalHomWithGivenObjects}(s, a, b, r).
‣ AddMorphismToLeftBidual( 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 MorphismToLeftBidual. F: ( a ) \mapsto \mathtt{MorphismToLeftBidual}(a).
‣ AddMorphismToLeftBidualWithGivenLeftBidual( 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 MorphismToLeftBidualWithGivenLeftBidual. F: ( a, r ) \mapsto \mathtt{MorphismToLeftBidualWithGivenLeftBidual}(a, r).
‣ AddTensorProductLeftDualityCompatibilityMorphism( 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 TensorProductLeftDualityCompatibilityMorphism. F: ( a, b ) \mapsto \mathtt{TensorProductLeftDualityCompatibilityMorphism}(a, b).
‣ AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects( 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 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{TensorProductLeftDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).
‣ AddTensorProductLeftInternalHomCompatibilityMorphism( 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 TensorProductLeftInternalHomCompatibilityMorphism. F: ( list ) \mapsto \mathtt{TensorProductLeftInternalHomCompatibilityMorphism}(list).
‣ AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects( 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 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects. F: ( source, list, range ) \mapsto \mathtt{TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range).
‣ AddTensorProductToLeftInternalHomAdjunctMorphism( 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 TensorProductToLeftInternalHomAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{TensorProductToLeftInternalHomAdjunctMorphism}(a, b, f).
‣ AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom( 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 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom}(a, b, f, i).
‣ AddUniversalPropertyOfLeftDual( 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 UniversalPropertyOfLeftDual. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfLeftDual}(t, a, alpha).
‣ AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit( 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 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit}(a).
‣ AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject( 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 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject}(a).
‣ AddIsomorphismFromLeftInternalCoHomToObject( 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 IsomorphismFromLeftInternalCoHomToObject. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomToObject}(a).
‣ AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom( 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 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom. F: ( a, s ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom}(a, s).
‣ AddIsomorphismFromObjectToLeftInternalCoHom( 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 IsomorphismFromObjectToLeftInternalCoHom. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalCoHom}(a).
‣ AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom( 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 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom}(a, r).
‣ AddLeftCoDualOnMorphisms( 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 LeftCoDualOnMorphisms. F: ( alpha ) \mapsto \mathtt{LeftCoDualOnMorphisms}(alpha).
‣ AddLeftCoDualOnMorphismsWithGivenLeftCoDuals( 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 LeftCoDualOnMorphismsWithGivenLeftCoDuals. F: ( s, alpha, r ) \mapsto \mathtt{LeftCoDualOnMorphismsWithGivenLeftCoDuals}(s, alpha, r).
‣ AddLeftCoDualOnObjects( 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 LeftCoDualOnObjects. F: ( a ) \mapsto \mathtt{LeftCoDualOnObjects}(a).
‣ AddLeftCoDualityTensorProductCompatibilityMorphism( 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 LeftCoDualityTensorProductCompatibilityMorphism. F: ( a, b ) \mapsto \mathtt{LeftCoDualityTensorProductCompatibilityMorphism}(a, b).
‣ AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects( 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 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r).
‣ AddLeftCoclosedMonoidalCoevaluationMorphism( 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 LeftCoclosedMonoidalCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{LeftCoclosedMonoidalCoevaluationMorphism}(a, b).
‣ AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource( 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 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource}(a, b, s).
‣ AddLeftCoclosedMonoidalEvaluationForLeftCoDual( 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 LeftCoclosedMonoidalEvaluationForLeftCoDual. F: ( a ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationForLeftCoDual}(a).
‣ AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct( 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 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct. F: ( s, a, r ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct}(s, a, r).
‣ AddLeftCoclosedMonoidalEvaluationMorphism( 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 LeftCoclosedMonoidalEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationMorphism}(a, b).
‣ AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange( 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 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationMorphismWithGivenRange}(a, b, r).
‣ AddLeftCoclosedMonoidalLambdaElimination( 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 LeftCoclosedMonoidalLambdaElimination. F: ( a, b, alpha ) \mapsto \mathtt{LeftCoclosedMonoidalLambdaElimination}(a, b, alpha).
‣ AddLeftCoclosedMonoidalLambdaIntroduction( 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 LeftCoclosedMonoidalLambdaIntroduction. F: ( alpha ) \mapsto \mathtt{LeftCoclosedMonoidalLambdaIntroduction}(alpha).
‣ AddLeftCoclosedMonoidalPostCoComposeMorphism( 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 LeftCoclosedMonoidalPostCoComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{LeftCoclosedMonoidalPostCoComposeMorphism}(a, b, c).
‣ AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects( 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 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddLeftCoclosedMonoidalPreCoComposeMorphism( 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 LeftCoclosedMonoidalPreCoComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{LeftCoclosedMonoidalPreCoComposeMorphism}(a, b, c).
‣ AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects( 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 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r).
‣ AddLeftInternalCoHomOnMorphisms( 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 LeftInternalCoHomOnMorphisms. F: ( alpha, beta ) \mapsto \mathtt{LeftInternalCoHomOnMorphisms}(alpha, beta).
‣ AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms( 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 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms. F: ( s, alpha, beta, r ) \mapsto \mathtt{LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms}(s, alpha, beta, r).
‣ AddLeftInternalCoHomOnObjects( 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 LeftInternalCoHomOnObjects. F: ( a, b ) \mapsto \mathtt{LeftInternalCoHomOnObjects}(a, b).
‣ AddLeftInternalCoHomTensorProductCompatibilityMorphism( 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 LeftInternalCoHomTensorProductCompatibilityMorphism. F: ( list ) \mapsto \mathtt{LeftInternalCoHomTensorProductCompatibilityMorphism}(list).
‣ AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects( 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 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects. F: ( source, list, range ) \mapsto \mathtt{LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range).
‣ AddLeftInternalCoHomToTensorProductAdjunctMorphism( 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 LeftInternalCoHomToTensorProductAdjunctMorphism. F: ( a, c, f ) \mapsto \mathtt{LeftInternalCoHomToTensorProductAdjunctMorphism}(a, c, f).
‣ AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct( 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 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct. F: ( a, c, f, t ) \mapsto \mathtt{LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct}(a, c, f, t).
‣ AddMorphismFromLeftCoBidual( 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 MorphismFromLeftCoBidual. F: ( a ) \mapsto \mathtt{MorphismFromLeftCoBidual}(a).
‣ AddMorphismFromLeftCoBidualWithGivenLeftCoBidual( 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 MorphismFromLeftCoBidualWithGivenLeftCoBidual. F: ( a, s ) \mapsto \mathtt{MorphismFromLeftCoBidualWithGivenLeftCoBidual}(a, s).
‣ AddMorphismFromLeftInternalCoHomToTensorProduct( 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 MorphismFromLeftInternalCoHomToTensorProduct. F: ( a, b ) \mapsto \mathtt{MorphismFromLeftInternalCoHomToTensorProduct}(a, b).
‣ AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects( 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 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r).
‣ AddTensorProductToLeftInternalCoHomAdjunctMorphism( 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 TensorProductToLeftInternalCoHomAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{TensorProductToLeftInternalCoHomAdjunctMorphism}(b, c, g).
‣ AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom( 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 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom}(b, c, g, i).
‣ AddUniversalPropertyOfLeftCoDual( 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 UniversalPropertyOfLeftCoDual. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfLeftCoDual}(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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