A \(6\)-tuple \(( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho )\) consisting of
a category \(\mathbf{C}\),
a functor \(\otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}\) compatible with the congruence of morphisms,
an object \(1 \in \mathbf{C}\),
a natural isomorphism \(\alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c\),
a natural isomorphism \(\lambda_{a}: 1 \otimes a \cong a\),
a natural isomorphism \(\rho_{a}: a \otimes 1 \cong a\),
is called a monoidal category, if
for all objects \(a,b,c,d\), the pentagon identity holds:
\((\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) \sim \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d}\),
for all objects \(a,c\), the triangle identity holds:
\(( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} \sim \mathrm{id}_a \otimes \lambda_c\).
The corresponding GAP property is given by IsMonoidalCategory.
‣ TensorProductOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes b, a' \otimes b')\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the tensor product \(\alpha \otimes \beta\).
‣ TensorProductOnMorphismsWithGivenTensorProducts( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes b, a' \otimes b')\)
The arguments are an object \(s = a \otimes b\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = a' \otimes b'\). The output is the tensor product \(\alpha \otimes \beta\).
‣ AssociatorRightToLeft( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\).
The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\).
‣ AssociatorRightToLeftWithGivenTensorProducts( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\).
The arguments are an object \(s = a \otimes (b \otimes c)\), three objects \(a,b,c\), and an object \(r = (a \otimes b) \otimes c\). The output is the associator \(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\).
‣ AssociatorLeftToRight( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\).
The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\).
‣ AssociatorLeftToRightWithGivenTensorProducts( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\).
The arguments are an object \(s = (a \otimes b) \otimes c\), three objects \(a,b,c\), and an object \(r = a \otimes (b \otimes c)\). The output is the associator \(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\).
‣ LeftUnitor( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1 \otimes a, a)\)
The argument is an object \(a\). The output is the left unitor \(\lambda_a: 1 \otimes a \rightarrow a\).
‣ LeftUnitorWithGivenTensorProduct( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(1 \otimes a, a)\)
The arguments are an object \(a\) and an object \(s = 1 \otimes a\). The output is the left unitor \(\lambda_a: 1 \otimes a \rightarrow a\).
‣ LeftUnitorInverse( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, 1 \otimes a)\)
The argument is an object \(a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \otimes a\).
‣ LeftUnitorInverseWithGivenTensorProduct( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, 1 \otimes a)\)
The argument is an object \(a\) and an object \(r = 1 \otimes a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \otimes a\).
‣ RightUnitor( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes 1, a)\)
The argument is an object \(a\). The output is the right unitor \(\rho_a: a \otimes 1 \rightarrow a\).
‣ RightUnitorWithGivenTensorProduct( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes 1, a)\)
The arguments are an object \(a\) and an object \(s = a \otimes 1\). The output is the right unitor \(\rho_a: a \otimes 1 \rightarrow a\).
‣ RightUnitorInverse( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, a \otimes 1)\)
The argument is an object \(a\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \otimes 1\).
‣ RightUnitorInverseWithGivenTensorProduct( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, a \otimes 1)\)
The arguments are an object \(a\) and an object \(r = a \otimes 1\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \otimes 1\).
‣ TensorProductOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects \(a, b\). The output is the tensor product \(a \otimes b\).
‣ AddTensorProductOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductOnObjects. \(F: (a,b) \mapsto a \otimes b\).
‣ TensorUnit( C ) | ( attribute ) |
Returns: an object
The argument is a category \(\mathbf{C}\). The output is the tensor unit \(1\) of \(\mathbf{C}\).
‣ AddTensorUnit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorUnit. \(F: ( ) \mapsto 1\).
‣ LeftDistributivityExpanding( a, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )\)
The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \(a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\).
‣ LeftDistributivityExpandingWithGivenObjects( s, a, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = a \otimes (b_1 \oplus \dots \oplus b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).
‣ LeftDistributivityFactoring( a, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )\)
The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \((a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n)\).
‣ LeftDistributivityFactoringWithGivenObjects( s, a, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = a \otimes (b_1 \oplus \dots \oplus b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).
‣ RightDistributivityExpanding( L, a ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )\)
The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\).
‣ RightDistributivityExpandingWithGivenObjects( s, L, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (b_1 \oplus \dots \oplus b_n) \otimes a\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\). The output is the right distributivity morphism \(s \rightarrow r\).
‣ RightDistributivityFactoring( L, a ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)\)
The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a \).
‣ RightDistributivityFactoringWithGivenObjects( s, L, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \oplus \dots \oplus b_n) \otimes a\). The output is the right distributivity morphism \(s \rightarrow r\).
A monoidal category \(\mathbf{C}\) equipped with a natural isomorphism \(B_{a,b}: a \otimes b \cong b \otimes a\) is called a braided monoidal category if
\(\lambda_a \circ B_{a,1} \sim \rho_a\),
\((B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} \sim \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c}\),
\(( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} \sim \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}\).
The corresponding GAP property is given by IsBraidedMonoidalCategory.
‣ Braiding( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \otimes b, b \otimes a )\).
The arguments are two objects \(a,b\). The output is the braiding \( B_{a,b}: a \otimes b \rightarrow b \otimes a\).
‣ BraidingWithGivenTensorProducts( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \otimes b, b \otimes a )\).
The arguments are an object \(s = a \otimes b\), two objects \(a,b\), and an object \(r = b \otimes a\). The output is the braiding \( B_{a,b}: a \otimes b \rightarrow b \otimes a\).
‣ BraidingInverse( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b \otimes a, a \otimes b )\).
The arguments are two objects \(a,b\). The output is the inverse braiding \( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\).
‣ BraidingInverseWithGivenTensorProducts( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b \otimes a, a \otimes b )\).
The arguments are an object \(s = b \otimes a\), two objects \(a,b\), and an object \(r = a \otimes b\). The output is the inverse braiding \( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\).
A braided monoidal category \(\mathbf{C}\) is called symmetric monoidal category if \(B_{a,b}^{-1} \sim B_{b,a}\). The corresponding GAP property is given by IsSymmetricMonoidalCategory.
A monoidal category \(\mathbf{C}\) which has for each functor \(- \otimes b: \mathbf{C} \rightarrow \mathbf{C}\) a right adjoint (denoted by \(\mathrm{\underline{Hom}}(b,-)\)) is called a closed monoidal category.
If no operations involving duals are installed manually, the dual objects will be derived as \(a^\vee \coloneqq \mathrm{\underline{Hom}}(a,1)\).
The corresponding GAP property is called IsClosedMonoidalCategory.
‣ InternalHomOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects \(a,b\). The output is the internal hom object \(\mathrm{\underline{Hom}}(a,b)\).
‣ InternalHomOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the internal hom morphism \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).
‣ InternalHomOnMorphismsWithGivenInternalHoms( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )\)
The arguments are an object \(s = \mathrm{\underline{Hom}}(a',b)\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = \mathrm{\underline{Hom}}(a,b')\). The output is the internal hom morphism \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).
‣ EvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )\).
The arguments are two objects \(a, b\). The output is the evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b\), i.e., the counit of the tensor hom adjunction.
‣ EvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )\).
The arguments are two objects \(a,b\) and an object \(s = \mathrm{\underline{Hom}}(a,b) \otimes a\). The output is the evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b\), i.e., the counit of the tensor hom adjunction.
‣ CoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )\).
The arguments are two objects \(a,b\). The output is the coevaluation morphism \(\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}}(b, a \otimes b)\), i.e., the unit of the tensor hom adjunction.
‣ CoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )\).
The arguments are two objects \(a,b\) and an object \(r = \mathrm{\underline{Hom}}(b, a \otimes b)\). The output is the coevaluation morphism \(\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}}(b, a \otimes b)\), i.e., the unit of the tensor hom adjunction.
‣ TensorProductToInternalHomAdjunctionMap( a, b, f ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) )\).
The arguments are two objects \(a,b\) and a morphism \(f: a \otimes b \rightarrow c\). The output is a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\) corresponding to \(f\) under the tensor hom adjunction.
‣ TensorProductToInternalHomAdjunctionMapWithGivenInternalHom( a, b, f, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) )\).
The arguments are two objects \(a,b\), a morphism \(f: a \otimes b \rightarrow c\) and an object \(i = \mathrm{\underline{Hom}}(b,c)\). The output is a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\) corresponding to \(f\) under the tensor hom adjunction.
‣ InternalHomToTensorProductAdjunctionMap( b, c, g ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes b, c)\).
The arguments are two objects \(b,c\) and a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\). The output is a morphism \(f: a \otimes b \rightarrow c\) corresponding to \(g\) under the tensor hom adjunction.
‣ InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct( b, c, g, t ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes b, c)\).
The arguments are two objects \(b,c\), a morphism \(g: a \rightarrow \mathrm{\underline{Hom}}(b,c)\) and an object \(t = a \otimes b\). The output is a morphism \(f: a \otimes b \rightarrow c\) corresponding to \(g\) under the tensor hom adjunction.
‣ MonoidalPreComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )\).
The arguments are three objects \(a,b,c\). The output is the precomposition morphism \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).
‣ MonoidalPreComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )\).
The arguments are an object \(s = \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{Hom}}(a,c)\). The output is the precomposition morphism \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).
‣ MonoidalPostComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )\).
The arguments are three objects \(a,b,c\). The output is the postcomposition morphism \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).
‣ MonoidalPostComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )\).
The arguments are an object \(s = \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{Hom}}(a,c)\). The output is the postcomposition morphism \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).
‣ DualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object \(a\). The output is its dual object \(a^{\vee}\).
‣ DualOnMorphisms( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( b^{\vee}, a^{\vee} )\).
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its dual morphism \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).
‣ DualOnMorphismsWithGivenDuals( s, alpha, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b^{\vee}, a^{\vee} )\).
The argument is an object \(s = b^{\vee}\), a morphism \(\alpha: a \rightarrow b\), and an object \(r = a^{\vee}\). The output is the dual morphism \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).
‣ EvaluationForDual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes a, 1 )\).
The argument is an object \(a\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\).
‣ EvaluationForDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes a, 1 )\).
The arguments are an object \(s = a^{\vee} \otimes a\), an object \(a\), and an object \(r = 1\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\).
‣ MorphismToBidual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, (a^{\vee})^{\vee})\).
The argument is an object \(a\). The output is the morphism to the bidual \(a \rightarrow (a^{\vee})^{\vee}\).
‣ MorphismToBidualWithGivenBidual( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, (a^{\vee})^{\vee})\).
The arguments are an object \(a\), and an object \(r = (a^{\vee})^{\vee}\). The output is the morphism to the bidual \(a \rightarrow (a^{\vee})^{\vee}\).
‣ TensorProductInternalHomCompatibilityMorphism( list ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))\).
The argument is a list of four objects \([ a, a', b, b' ]\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).
‣ TensorProductInternalHomCompatibilityMorphismWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))\).
The arguments are a list of four objects \([ a, a', b, b' ]\), and two objects \(s = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\) and \(r = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).
‣ TensorProductDualityCompatibilityMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).
‣ TensorProductDualityCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )\).
The arguments are an object \(s = a^{\vee} \otimes b^{\vee}\), two objects \(a,b\), and an object \(r = (a \otimes b)^{\vee}\). The output is the natural morphism \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).
‣ MorphismFromTensorProductToInternalHom( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).
‣ MorphismFromTensorProductToInternalHomWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).
The arguments are an object \(s = a^{\vee} \otimes b\), two objects \(a,b\), and an object \(r = \mathrm{\underline{Hom}}(a,b)\). The output is the natural morphism \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).
‣ IsomorphismFromDualObjectToInternalHomIntoTensorUnit( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}}(a,1))\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1)\).
‣ IsomorphismFromInternalHomIntoTensorUnitToDualObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(a,1), a^{\vee})\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}\).
‣ UniversalPropertyOfDual( t, a, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(t, a^{\vee})\).
The arguments are two objects \(t,a\), and a morphism \(\alpha: t \otimes a \rightarrow 1\). The output is the morphism \(t \rightarrow a^{\vee}\) given by the universal property of \(a^{\vee}\).
‣ LambdaIntroduction( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) )\).
The argument is a morphism \(\alpha: a \rightarrow b\). The output is the corresponding morphism \(1 \rightarrow \mathrm{\underline{Hom}}(a,b)\) under the tensor hom adjunction.
‣ LambdaElimination( a, b, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\).
The arguments are two objects \(a,b\), and a morphism \(\alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b)\). The output is a morphism \(a \rightarrow b\) corresponding to \(\alpha\) under the tensor hom adjunction.
‣ IsomorphismFromObjectToInternalHom( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))\).
The argument is an object \(a\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{Hom}}(1,a)\).
‣ IsomorphismFromObjectToInternalHomWithGivenInternalHom( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))\).
The argument is an object \(a\), and an object \(r = \mathrm{\underline{Hom}}(1,a)\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{Hom}}(1,a)\).
‣ IsomorphismFromInternalHomToObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)\).
The argument is an object \(a\). The output is the natural isomorphism \(\mathrm{\underline{Hom}}(1,a) \rightarrow a\).
‣ IsomorphismFromInternalHomToObjectWithGivenInternalHom( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)\).
The argument is an object \(a\), and an object \(s = \mathrm{\underline{Hom}}(1,a)\). The output is the natural isomorphism \(\mathrm{\underline{Hom}}(1,a) \rightarrow a\).
A monoidal category \(\mathbf{C}\) which has for each functor \(- \otimes b: \mathbf{C} \rightarrow \mathbf{C}\) a left adjoint (denoted by \(\mathrm{\underline{coHom}}(-,b)\)) is called a coclosed monoidal category.
If no operations involving coduals are installed manually, the codual objects will be derived as \(a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a)\).
The corresponding GAP property is called IsCoclosedMonoidalCategory.
‣ InternalCoHomOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects \(a,b\). The output is the internal cohom object \(\mathrm{\underline{coHom}}(a,b)\).
‣ InternalCoHomOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the internal cohom morphism \(\mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b)\).
‣ InternalCoHomOnMorphismsWithGivenInternalCoHoms( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )\)
The arguments are an object \(s = \mathrm{\underline{coHom}}(a,b')\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = \mathrm{\underline{coHom}}(a',b)\). The output is the internal cohom morphism \(\mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b)\).
‣ CoclosedEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b )\).
The arguments are two objects \(a, b\). The output is the coclosed evaluation morphism \(\mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b\), i.e., the unit of the cohom tensor adjunction.
‣ CoclosedEvaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b )\).
The arguments are two objects \(a,b\) and an object \(r = \mathrm{\underline{coHom}}(a,b) \otimes b\). The output is the coclosed evaluation morphism \(\mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b\), i.e., the unit of the cohom tensor adjunction.
‣ CoclosedCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), a )\).
The arguments are two objects \(a,b\). The output is the coclosed coevaluation morphism \(\mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a\), i.e., the counit of the cohom tensor adjunction.
‣ CoclosedCoevaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), b )\).
The arguments are two objects \(a,b\) and an object \(s = \mathrm{\underline{coHom}(a \otimes b, b)}\). The output is the coclosed coevaluation morphism \(\mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a\), i.e., the unit of the cohom tensor adjunction.
‣ TensorProductToInternalCoHomAdjunctionMap( c, b, g ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c )\).
The arguments are two objects \(c,b\) and a morphism \(g: a \rightarrow c \otimes b\). The output is a morphism \(f: \mathrm{\underline{coHom}}(a,b) \rightarrow c\) corresponding to \(g\) under the cohom tensor adjunction.
‣ TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom( c, b, g, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c )\).
The arguments are two objects \(c,b\), a morphism \(g: a \rightarrow c \otimes b\) and an object \(i = \mathrm{\underline{coHom}(a,b)}\). The output is a morphism \(f: \mathrm{\underline{coHom}}(a,b) \rightarrow c\) corresponding to \(g\) under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductAdjunctionMap( a, b, f ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, c \otimes b)\).
The arguments are two objects \(a,b\) and a morphism \(f: \mathrm{\underline{coHom}}(a,b) \rightarrow c\). The output is a morphism \(g: a \rightarrow c \otimes b\) corresponding to \(f\) under the cohom tensor adjunction.
‣ InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct( a, b, f, t ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, c \otimes b)\).
The arguments are two objects \(a,b\), a morphism \(f: \mathrm{\underline{coHom}}(a,b) \rightarrow c\) and an object \(t = c \otimes b\). The output is a morphism \(g: a \rightarrow c \otimes b\) corresponding to \(f\) under the cohom tensor adjunction.
‣ MonoidalPreCoComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) )\).
The arguments are three objects \(a,b,c\). The output is the precocomposition morphism \(\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b)\).
‣ MonoidalPreCoComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) )\).
The arguments are an object \(s = \mathrm{\underline{coHom}}(a,c)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c)\). The output is the precocomposition morphism \(\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b)\).
‣ MonoidalPostCoComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) )\).
The arguments are three objects \(a,b,c\). The output is the postcocomposition morphism \(\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c)\).
‣ MonoidalPostCoComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) )\).
The arguments are an object \(s = \mathrm{\underline{coHom}}(a,c)\), three objects \(a,b,c\), and an object \(r = \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b)\). The output is the postcocomposition morphism \(\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c)\).
‣ CoDualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object \(a\). The output is its codual object \(a_{\vee}\).
‣ CoDualOnMorphisms( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( b_{\vee}, a_{\vee} )\).
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its codual morphism \(\alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}\).
‣ CoDualOnMorphismsWithGivenCoDuals( s, alpha, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b_{\vee}, a_{\vee} )\).
The argument is an object \(s = b_{\vee}\), a morphism \(\alpha: a \rightarrow b\), and an object \(r = a_{\vee}\). The output is the dual morphism \(\alpha_{\vee}: b^{\vee} \rightarrow a^{\vee}\).
‣ CoclosedEvaluationForCoDual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( 1, a_{\vee} \otimes a )\).
The argument is an object \(a\). The output is the coclosed evaluation morphism \(\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a\).
‣ CoclosedEvaluationForCoDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( 1, a_{\vee} \otimes a )\).
The arguments are an object \(s = 1\), an object \(a\), and an object \(r = a_{\vee} \otimes a\). The output is the coclosed evaluation morphism \(\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a\).
‣ MorphismFromCoBidual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}((a_{\vee})_{\vee}, a)\).
The argument is an object \(a\). The output is the morphism from the cobidual \((a_{\vee})_{\vee} \rightarrow a\).
‣ MorphismFromCoBidualWithGivenCoBidual( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}((a_{\vee})_{\vee}, a)\).
The arguments are an object \(a\), and an object \(s = (a_{\vee})_{\vee}\). The output is the morphism from the cobidual \((a_{\vee})_{\vee} \rightarrow a\).
‣ InternalCoHomTensorProductCompatibilityMorphism( list ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'))\).
The argument is a list of four objects \([ a, a', b, b' ]\). The output is the natural morphism \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\).
‣ InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') )\).
The arguments are a list of four objects \([ a, a', b, b' ]\), and two objects \(s = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\) and \(r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\). The output is the natural morphism \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\).
‣ CoDualityTensorProductCompatibilityMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}\).
‣ CoDualityTensorProductCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} )\).
The arguments are an object \(s = (a \otimes b)_{\vee}\), two objects \(a,b\), and an object \(r = a_{\vee} \otimes b_{\vee}\). The output is the natural morphism \(\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}\).
‣ MorphismFromInternalCoHomToTensorProduct( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a\).
‣ MorphismFromInternalCoHomToTensorProductWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), a \otimes b_{\vee} )\).
The arguments are an object \(s = \mathrm{\underline{coHom}}(a,b)\), two objects \(a,b\), and an object \(r = b_{\vee} \otimes a\). The output is the natural morphism \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}\).
‣ IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}}(1,a))\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a)\).
‣ IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{coHom}}(1,a), a_{\vee})\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}: \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}\).
‣ UniversalPropertyOfCoDual( t, a, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a_{\vee}, t)\).
The arguments are two objects \(t,a\), and a morphism \(\alpha: 1 \rightarrow t \otimes a\). The output is the morphism \(a_{\vee} \rightarrow t\) given by the universal property of \(a_{\vee}\).
‣ CoLambdaIntroduction( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), 1 )\).
The argument is a morphism \(\alpha: a \rightarrow b\). The output is the corresponding morphism \( \mathrm{\underline{coHom}}(a,b) \rightarrow 1\) under the cohom tensor adjunction.
‣ CoLambdaElimination( a, b, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\).
The arguments are two objects \(a,b\), and a morphism \(\alpha: \mathrm{\underline{coHom}}(a,b) \rightarrow 1\). The output is a morphism \(a \rightarrow b\) corresponding to \(\alpha\) under the cohom tensor adjunction.
‣ IsomorphismFromObjectToInternalCoHom( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1))\).
The argument is an object \(a\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{coHom}}(a,1)\).
‣ IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1))\).
The argument is an object \(a\), and an object \(r = \mathrm{\underline{coHom}}(a,1)\). The output is the natural isomorphism \(a \rightarrow \mathrm{\underline{coHom}}(a,1)\).
‣ IsomorphismFromInternalCoHomToObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a)\).
The argument is an object \(a\). The output is the natural isomorphism \(\mathrm{\underline{coHom}}(a,1) \rightarrow a\).
‣ IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a)\).
The argument is an object \(a\), and an object \(s = \mathrm{\underline{coHom}}(a,1)\). The output is the natural isomorphism \(\mathrm{\underline{coHom}}(a,1) \rightarrow a\).
A monoidal category \(\mathbf{C}\) which is symmetric and closed is called a symmetric closed monoidal category.
The corresponding GAP property is given by IsSymmetricClosedMonoidalCategory.
A monoidal category \(\mathbf{C}\) which is symmetric and coclosed is called a symmetric coclosed monoidal category.
The corresponding GAP property is given by IsSymmetricCoclosedMonoidalCategory.
A symmetric closed monoidal category \(\mathbf{C}\) satisfying
the natural morphism
\(\mathrm{\underline{Hom}}(a, a') \otimes \mathrm{\underline{Hom}}(b, b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b, a' \otimes b')\) is an isomorphism,
the natural morphism
\(a \rightarrow \mathrm{\underline{Hom}}(\mathrm{\underline{Hom}}(a, 1), 1)\) is an isomorphism is called a rigid symmetric closed monoidal category.
If no operations involving the closed structure are installed manually, the internal hom objects will be derived as \(\mathrm{\underline{Hom}}(a,b) \coloneqq a^\vee \otimes b\) and, in particular, \(\mathrm{\underline{Hom}}(a,1) \coloneqq a^\vee \otimes 1\).
The corresponding GAP property is given by IsRigidSymmetricClosedMonoidalCategory.
‣ IsomorphismFromTensorProductWithDualObjectToInternalHom( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).
‣ IsomorphismFromInternalHomToTensorProductWithDualObject( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).
The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}\), namely \(\mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).
‣ MorphismFromInternalHomToTensorProduct( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).
The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}\), namely \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).
‣ MorphismFromInternalHomToTensorProductWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )\).
The arguments are an object \(s = \mathrm{\underline{Hom}}(a,b)\), two objects \(a,b\), and an object \(r = a^{\vee} \otimes b\). The output is the inverse of \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}\), namely \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).
‣ TensorProductInternalHomCompatibilityMorphismInverse( list ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') )\).
The argument is a list of four objects \([ a, a', b, b' ]\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).
‣ TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') )\).
The arguments are a list of four objects \([ a, a', b, b' ]\), and two objects \(s = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\) and \(r = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\). The output is the natural morphism \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).
‣ CoevaluationForDual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1,a \otimes a^{\vee})\).
The argument is an object \(a\). The output is the coevaluation morphism \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).
‣ CoevaluationForDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(1,a \otimes a^{\vee})\).
The arguments are an object \(s = 1\), an object \(a\), and an object \(r = a \otimes a^{\vee}\). The output is the coevaluation morphism \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).
‣ TraceMap( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1,1)\).
The argument is an endomorphism \(\alpha: a \rightarrow a\). The output is the trace morphism \(\mathrm{trace}_{\alpha}: 1 \rightarrow 1\).
‣ RankMorphism( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1,1)\).
The argument is an object \(a\). The output is the rank morphism \(\mathrm{rank}_a: 1 \rightarrow 1\).
‣ MorphismFromBidual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}((a^{\vee})^{\vee},a)\).
The argument is an object \(a\). The output is the inverse of the morphism to the bidual \((a^{\vee})^{\vee} \rightarrow a\).
‣ MorphismFromBidualWithGivenBidual( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}((a^{\vee})^{\vee},a)\).
The argument is an object \(a\), and an object \(s = (a^{\vee})^{\vee}\). The output is the inverse of the morphism to the bidual \((a^{\vee})^{\vee} \rightarrow a\).
A symmetric coclosed monoidal category \(\mathbf{C}\) satisfying
the natural morphism
\(\mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a, b) \otimes \mathrm{\underline{coHom}}(a', b')\) is an isomorphism,
the natural morphism
\(\mathrm{\underline{coHom}}(1, \mathrm{\underline{coHom}}(1, a)) \rightarrow a\) is an isomorphism is called a rigid symmetric coclosed monoidal category.
If no operations involving the coclosed structure are installed manually, the internal cohom objects will be derived as \(\mathrm{\underline{coHom}}(a,b) \coloneqq a \otimes b_\vee\) and, in particular, \(\mathrm{\underline{coHom}}(1,a) \coloneqq 1 \otimes a_\vee\).
The corresponding GAP property is given by IsRigidSymmetricCoclosedMonoidalCategory.
‣ IsomorphismFromInternalCoHomToTensorProductWithCoDualObject( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a\).
‣ IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a)\).
The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}\), namely \(\mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).
‣ MorphismFromTensorProductToInternalCoHom( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a) )\).
The arguments are two objects \(a,b\). The output is the inverse of \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}\), namely \(\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).
‣ MorphismFromTensorProductToInternalCoHomWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a)\).
The arguments are an object \(s_{\vee} = a \otimes b\), two objects \(a,b\), and an object \(r = \mathrm{\underline{coHom}}(b,a)\). The output is the inverse of \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}\), namely \(\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).
‣ InternalCoHomTensorProductCompatibilityMorphismInverse( list ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' )\).
The argument is a list of four objects \([ a, a', b, b' ]\). The output is the natural morphism \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\).
‣ InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' )\).
The arguments are a list of four objects \([ a, a', b, b' ]\), and two objects \(s = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\) and \(r = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\). The output is the natural morphism \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\).
‣ CoclosedCoevaluationForCoDual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes a_{\vee}, 1)\).
The argument is an object \(a\). The output is the coclosed coevaluation morphism \(\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1\).
‣ CoclosedCoevaluationForCoDualWithGivenTensorProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \otimes a_{\vee}, 1)\).
The arguments are an object \(s = a \otimes a_{\vee}\), an object \(a\), and an object \(r = 1\). The output is the coclosed coevaluation morphism \(\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1\).
‣ CoTraceMap( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1,1)\).
The argument is an endomorphism \(\alpha: a \rightarrow a\). The output is the cotrace morphism \(\mathrm{cotrace}_{\alpha}: 1 \rightarrow 1\).
‣ CoRankMorphism( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1,1)\).
The argument is an object \(a\). The output is the corank morphism \(\mathrm{corank}_a: 1 \rightarrow 1\).
‣ MorphismToCoBidual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, (a_{\vee})_{\vee})\).
The argument is an object \(a\). The output is the inverse of the morphism from the cobidual \(a \rightarrow (a_{\vee})_{\vee}\).
‣ MorphismToCoBidualWithGivenCoBidual( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,(a_{\vee})_{\vee})\).
The argument is an object \(a\), and an object \(r = (a_{\vee})_{\vee}\). The output is the inverse of the morphism from the cobidual \(a \rightarrow (a_{\vee})_{\vee}\).
‣ InternalHom( a, b ) | ( operation ) |
Returns: a cell
This is a convenience method. The arguments are two cells \(a,b\). The output is the internal hom cell. If \(a,b\) are two CAP objects the output is the internal Hom object \(\mathrm{\underline{Hom}}(a,b)\). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal hom on morphisms, where any object is replaced by its identity morphism.
‣ InternalCoHom( a, b ) | ( operation ) |
Returns: a cell
This is a convenience method. The arguments are two cells \(a,b\). The output is the internal cohom cell. If \(a,b\) are two CAP objects the output is the internal cohom object \(\mathrm{\underline{coHom}}(a,b)\). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal cohom on morphisms, where any object is replaced by its identity morphism.
‣ AddLeftDistributivityExpanding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftDistributivityExpanding. \(F: ( a, L ) \mapsto \mathtt{LeftDistributivityExpanding}(a, L)\).
‣ AddLeftDistributivityExpandingWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftDistributivityExpandingWithGivenObjects. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r)\).
‣ AddLeftDistributivityFactoring( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftDistributivityFactoring. \(F: ( a, L ) \mapsto \mathtt{LeftDistributivityFactoring}(a, L)\).
‣ AddLeftDistributivityFactoringWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftDistributivityFactoringWithGivenObjects. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r)\).
‣ AddRightDistributivityExpanding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightDistributivityExpanding. \(F: ( L, a ) \mapsto \mathtt{RightDistributivityExpanding}(L, a)\).
‣ AddRightDistributivityExpandingWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightDistributivityExpandingWithGivenObjects. \(F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r)\).
‣ AddRightDistributivityFactoring( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightDistributivityFactoring. \(F: ( L, a ) \mapsto \mathtt{RightDistributivityFactoring}(L, a)\).
‣ AddRightDistributivityFactoringWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightDistributivityFactoringWithGivenObjects. \(F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r)\).
‣ AddBraiding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Braiding. \(F: ( a, b ) \mapsto \mathtt{Braiding}(a, b)\).
‣ AddBraidingInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BraidingInverse. \(F: ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b)\).
‣ AddBraidingInverseWithGivenTensorProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BraidingInverseWithGivenTensorProducts. \(F: ( s, a, b, r ) \mapsto \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r)\).
‣ AddBraidingWithGivenTensorProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BraidingWithGivenTensorProducts. \(F: ( s, a, b, r ) \mapsto \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r)\).
‣ AddCoevaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoevaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CoevaluationMorphism}(a, b)\).
‣ AddCoevaluationMorphismWithGivenRange( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoevaluationMorphismWithGivenRange. \(F: ( a, b, r ) \mapsto \mathtt{CoevaluationMorphismWithGivenRange}(a, b, r)\).
‣ AddDualOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DualOnMorphisms. \(F: ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha)\).
‣ AddDualOnMorphismsWithGivenDuals( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DualOnMorphismsWithGivenDuals. \(F: ( s, alpha, r ) \mapsto \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r)\).
‣ AddDualOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DualOnObjects. \(F: ( a ) \mapsto \mathtt{DualOnObjects}(a)\).
‣ AddEvaluationForDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EvaluationForDual. \(F: ( a ) \mapsto \mathtt{EvaluationForDual}(a)\).
‣ AddEvaluationForDualWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EvaluationForDualWithGivenTensorProduct. \(F: ( s, a, r ) \mapsto \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r)\).
‣ AddEvaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EvaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{EvaluationMorphism}(a, b)\).
‣ AddEvaluationMorphismWithGivenSource( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EvaluationMorphismWithGivenSource. \(F: ( a, b, s ) \mapsto \mathtt{EvaluationMorphismWithGivenSource}(a, b, s)\).
‣ AddInternalHomOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalHomOnMorphisms. \(F: ( alpha, beta ) \mapsto \mathtt{InternalHomOnMorphisms}(alpha, beta)\).
‣ AddInternalHomOnMorphismsWithGivenInternalHoms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalHomOnMorphismsWithGivenInternalHoms. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta, r)\).
‣ AddInternalHomOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalHomOnObjects. \(F: ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a, b)\).
‣ AddInternalHomToTensorProductAdjunctionMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalHomToTensorProductAdjunctionMap. \(F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMap}(b, c, g)\).
‣ AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct. \(F: ( b, c, g, t ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct}(b, c, g, t)\).
‣ AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromDualObjectToInternalHomIntoTensorUnit. \(F: ( a ) \mapsto \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a)\).
‣ AddIsomorphismFromInternalHomIntoTensorUnitToDualObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomIntoTensorUnitToDualObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a)\).
‣ AddIsomorphismFromInternalHomToObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomToObject}(a)\).
‣ AddIsomorphismFromInternalHomToObjectWithGivenInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToObjectWithGivenInternalHom. \(F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s)\).
‣ AddIsomorphismFromObjectToInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToInternalHom. \(F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalHom}(a)\).
‣ AddIsomorphismFromObjectToInternalHomWithGivenInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToInternalHomWithGivenInternalHom. \(F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r)\).
‣ AddLambdaElimination( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LambdaElimination. \(F: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha)\).
‣ AddLambdaIntroduction( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LambdaIntroduction. \(F: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha)\).
‣ AddMonoidalPostComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPostComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPostComposeMorphism}(a, b, c)\).
‣ AddMonoidalPostComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPostComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddMonoidalPreComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPreComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPreComposeMorphism}(a, b, c)\).
‣ AddMonoidalPreComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPreComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddMorphismFromTensorProductToInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromTensorProductToInternalHom. \(F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalHom}(a, b)\).
‣ AddMorphismFromTensorProductToInternalHomWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromTensorProductToInternalHomWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s, a, b, r)\).
‣ AddMorphismToBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToBidual. \(F: ( a ) \mapsto \mathtt{MorphismToBidual}(a)\).
‣ AddMorphismToBidualWithGivenBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToBidualWithGivenBidual. \(F: ( a, r ) \mapsto \mathtt{MorphismToBidualWithGivenBidual}(a, r)\).
‣ AddTensorProductDualityCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductDualityCompatibilityMorphism. \(F: ( a, b ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphism}(a, b)\).
‣ AddTensorProductDualityCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductDualityCompatibilityMorphismWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r)\).
‣ AddTensorProductInternalHomCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphism. \(F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphism}(list)\).
‣ AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphismWithGivenObjects. \(F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range)\).
‣ AddTensorProductToInternalHomAdjunctionMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductToInternalHomAdjunctionMap. \(F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMap}(a, b, f)\).
‣ AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductToInternalHomAdjunctionMapWithGivenInternalHom. \(F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMapWithGivenInternalHom}(a, b, f, i)\).
‣ AddUniversalPropertyOfDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalPropertyOfDual. \(F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfDual}(t, a, alpha)\).
‣ AddCoDualOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoDualOnMorphisms. \(F: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha)\).
‣ AddCoDualOnMorphismsWithGivenCoDuals( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoDualOnMorphismsWithGivenCoDuals. \(F: ( s, alpha, r ) \mapsto \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r)\).
‣ AddCoDualOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoDualOnObjects. \(F: ( a ) \mapsto \mathtt{CoDualOnObjects}(a)\).
‣ AddCoDualityTensorProductCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoDualityTensorProductCompatibilityMorphism. \(F: ( a, b ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b)\).
‣ AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoDualityTensorProductCompatibilityMorphismWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r)\).
‣ AddCoLambdaElimination( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoLambdaElimination. \(F: ( a, b, alpha ) \mapsto \mathtt{CoLambdaElimination}(a, b, alpha)\).
‣ AddCoLambdaIntroduction( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoLambdaIntroduction. \(F: ( alpha ) \mapsto \mathtt{CoLambdaIntroduction}(alpha)\).
‣ AddCoclosedCoevaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedCoevaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CoclosedCoevaluationMorphism}(a, b)\).
‣ AddCoclosedCoevaluationMorphismWithGivenSource( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedCoevaluationMorphismWithGivenSource. \(F: ( a, b, s ) \mapsto \mathtt{CoclosedCoevaluationMorphismWithGivenSource}(a, b, s)\).
‣ AddCoclosedEvaluationForCoDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedEvaluationForCoDual. \(F: ( a ) \mapsto \mathtt{CoclosedEvaluationForCoDual}(a)\).
‣ AddCoclosedEvaluationForCoDualWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedEvaluationForCoDualWithGivenTensorProduct. \(F: ( s, a, r ) \mapsto \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r)\).
‣ AddCoclosedEvaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedEvaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CoclosedEvaluationMorphism}(a, b)\).
‣ AddCoclosedEvaluationMorphismWithGivenRange( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedEvaluationMorphismWithGivenRange. \(F: ( a, b, r ) \mapsto \mathtt{CoclosedEvaluationMorphismWithGivenRange}(a, b, r)\).
‣ AddInternalCoHomOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomOnMorphisms. \(F: ( alpha, beta ) \mapsto \mathtt{InternalCoHomOnMorphisms}(alpha, beta)\).
‣ AddInternalCoHomOnMorphismsWithGivenInternalCoHoms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomOnMorphismsWithGivenInternalCoHoms. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s, alpha, beta, r)\).
‣ AddInternalCoHomOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomOnObjects. \(F: ( a, b ) \mapsto \mathtt{InternalCoHomOnObjects}(a, b)\).
‣ AddInternalCoHomTensorProductCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphism. \(F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list)\).
‣ AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects. \(F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range)\).
‣ AddInternalCoHomToTensorProductAdjunctionMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomToTensorProductAdjunctionMap. \(F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMap}(a, b, f)\).
‣ AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct. \(F: ( a, b, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct}(a, b, f, t)\).
‣ AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit. \(F: ( a ) \mapsto \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a)\).
‣ AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a)\).
‣ AddIsomorphismFromInternalCoHomToObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalCoHomToObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObject}(a)\).
‣ AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom. \(F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s)\).
‣ AddIsomorphismFromObjectToInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToInternalCoHom. \(F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHom}(a)\).
‣ AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom. \(F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r)\).
‣ AddMonoidalPostCoComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPostCoComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPostCoComposeMorphism}(a, b, c)\).
‣ AddMonoidalPostCoComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPostCoComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddMonoidalPreCoComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPreCoComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPreCoComposeMorphism}(a, b, c)\).
‣ AddMonoidalPreCoComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MonoidalPreCoComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddMorphismFromCoBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromCoBidual. \(F: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a)\).
‣ AddMorphismFromCoBidualWithGivenCoBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromCoBidualWithGivenCoBidual. \(F: ( a, s ) \mapsto \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s)\).
‣ AddMorphismFromInternalCoHomToTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromInternalCoHomToTensorProduct. \(F: ( a, b ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b)\).
‣ AddMorphismFromInternalCoHomToTensorProductWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromInternalCoHomToTensorProductWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r)\).
‣ AddTensorProductToInternalCoHomAdjunctionMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductToInternalCoHomAdjunctionMap. \(F: ( c, b, g ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMap}(c, b, g)\).
‣ AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom. \(F: ( c, b, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom}(c, b, g, i)\).
‣ AddUniversalPropertyOfCoDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalPropertyOfCoDual. \(F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCoDual}(t, a, alpha)\).
‣ AddAssociatorLeftToRight( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AssociatorLeftToRight. \(F: ( a, b, c ) \mapsto \mathtt{AssociatorLeftToRight}(a, b, c)\).
‣ AddAssociatorLeftToRightWithGivenTensorProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AssociatorLeftToRightWithGivenTensorProducts. \(F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r)\).
‣ AddAssociatorRightToLeft( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AssociatorRightToLeft. \(F: ( a, b, c ) \mapsto \mathtt{AssociatorRightToLeft}(a, b, c)\).
‣ AddAssociatorRightToLeftWithGivenTensorProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AssociatorRightToLeftWithGivenTensorProducts. \(F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r)\).
‣ AddLeftUnitor( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftUnitor. \(F: ( a ) \mapsto \mathtt{LeftUnitor}(a)\).
‣ AddLeftUnitorInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftUnitorInverse. \(F: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a)\).
‣ AddLeftUnitorInverseWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftUnitorInverseWithGivenTensorProduct. \(F: ( a, r ) \mapsto \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r)\).
‣ AddLeftUnitorWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftUnitorWithGivenTensorProduct. \(F: ( a, s ) \mapsto \mathtt{LeftUnitorWithGivenTensorProduct}(a, s)\).
‣ AddRightUnitor( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightUnitor. \(F: ( a ) \mapsto \mathtt{RightUnitor}(a)\).
‣ AddRightUnitorInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightUnitorInverse. \(F: ( a ) \mapsto \mathtt{RightUnitorInverse}(a)\).
‣ AddRightUnitorInverseWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightUnitorInverseWithGivenTensorProduct. \(F: ( a, r ) \mapsto \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r)\).
‣ AddRightUnitorWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightUnitorWithGivenTensorProduct. \(F: ( a, s ) \mapsto \mathtt{RightUnitorWithGivenTensorProduct}(a, s)\).
‣ AddTensorProductOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductOnMorphisms. \(F: ( alpha, beta ) \mapsto \mathtt{TensorProductOnMorphisms}(alpha, beta)\).
‣ AddTensorProductOnMorphismsWithGivenTensorProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductOnMorphismsWithGivenTensorProducts. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s, alpha, beta, r)\).
‣ AddCoevaluationForDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoevaluationForDual. \(F: ( a ) \mapsto \mathtt{CoevaluationForDual}(a)\).
‣ AddCoevaluationForDualWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoevaluationForDualWithGivenTensorProduct. \(F: ( s, a, r ) \mapsto \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r)\).
‣ AddIsomorphismFromInternalHomToTensorProductWithDualObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalHomToTensorProductWithDualObject. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b)\).
‣ AddIsomorphismFromTensorProductWithDualObjectToInternalHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromTensorProductWithDualObjectToInternalHom. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b)\).
‣ AddMorphismFromBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromBidual. \(F: ( a ) \mapsto \mathtt{MorphismFromBidual}(a)\).
‣ AddMorphismFromBidualWithGivenBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromBidualWithGivenBidual. \(F: ( a, s ) \mapsto \mathtt{MorphismFromBidualWithGivenBidual}(a, s)\).
‣ AddMorphismFromInternalHomToTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromInternalHomToTensorProduct. \(F: ( a, b ) \mapsto \mathtt{MorphismFromInternalHomToTensorProduct}(a, b)\).
‣ AddMorphismFromInternalHomToTensorProductWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromInternalHomToTensorProductWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s, a, b, r)\).
‣ AddRankMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RankMorphism. \(F: ( a ) \mapsto \mathtt{RankMorphism}(a)\).
‣ AddTensorProductInternalHomCompatibilityMorphismInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphismInverse. \(F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list)\).
‣ AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects. \(F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source, list, range)\).
‣ AddTraceMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TraceMap. \(F: ( alpha ) \mapsto \mathtt{TraceMap}(alpha)\).
‣ AddCoRankMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoRankMorphism. \(F: ( a ) \mapsto \mathtt{CoRankMorphism}(a)\).
‣ AddCoTraceMap( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoTraceMap. \(F: ( alpha ) \mapsto \mathtt{CoTraceMap}(alpha)\).
‣ AddCoclosedCoevaluationForCoDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedCoevaluationForCoDual. \(F: ( a ) \mapsto \mathtt{CoclosedCoevaluationForCoDual}(a)\).
‣ AddCoclosedCoevaluationForCoDualWithGivenTensorProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoclosedCoevaluationForCoDualWithGivenTensorProduct. \(F: ( s, a, r ) \mapsto \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r)\).
‣ AddInternalCoHomTensorProductCompatibilityMorphismInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismInverse. \(F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list)\).
‣ AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects. \(F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source, list, range)\).
‣ AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInternalCoHomToTensorProductWithCoDualObject. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b)\).
‣ AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b)\).
‣ AddMorphismFromTensorProductToInternalCoHom( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromTensorProductToInternalCoHom. \(F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b)\).
‣ AddMorphismFromTensorProductToInternalCoHomWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromTensorProductToInternalCoHomWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s, a, b, r)\).
‣ AddMorphismToCoBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToCoBidual. \(F: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a)\).
‣ AddMorphismToCoBidualWithGivenCoBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToCoBidualWithGivenCoBidual. \(F: ( a, r ) \mapsto \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r)\).
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