A \(6\)-tuple \(( \mathbf{C}, \times, 1, \alpha, \lambda, \rho )\) consisting of
a category \(\mathbf{C}\),
a functor \(\times: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}\),
an object \(1 \in \mathbf{C}\),
a natural isomorphism \(\alpha_{a,b,c}: a \times (b \times c) \cong (a \times b) \times c\),
a natural isomorphism \(\lambda_{a}: 1 \times a \cong a\),
a natural isomorphism \(\rho_{a}: a \times 1 \cong a\),
is called a cartesian category, if
for all objects \(a,b,c,d\), the pentagon identity holds:
\((\alpha_{a,b,c} \times \mathrm{id}_d) \circ \alpha_{a,b \times c, d} \circ ( \mathrm{id}_a \times \alpha_{b,c,d} ) = \alpha_{a \times b, c, d} \circ \alpha_{a,b,c \times d}\),
for all objects \(a,c\), the triangle identity holds:
\(( \rho_a \times \mathrm{id}_c ) \circ \alpha_{a,1,c} = \mathrm{id}_a \times \lambda_c\).
The corresponding GAP property is given by IsCartesianCategory.
‣ CartesianBraiding( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \times b, b \times a )\).
The arguments are two objects \(a,b\). The output is the braiding \( B_{a,b}: a \times b \rightarrow b \times a\).
‣ CartesianBraidingWithGivenDirectProducts( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \times b, b \times a )\).
The arguments are an object \(s = a \times b\), two objects \(a,b\), and an object \(r = b \times a\). The output is the braiding \( B_{a,b}: a \times b \rightarrow b \times a\).
‣ CartesianBraidingInverse( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b \times a, a \times b )\).
The arguments are two objects \(a,b\). The output is the inverse braiding \( B_{a,b}^{-1}: b \times a \rightarrow a \times b\).
‣ CartesianBraidingInverseWithGivenDirectProducts( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b \times a, a \times b )\).
The arguments are an object \(s = b \times a\), two objects \(a,b\), and an object \(r = a \times b\). The output is the inverse braiding \( B_{a,b}^{-1}: b \times a \rightarrow a \times b\).
‣ DirectProductOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times b, a' \times b')\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the direct product \(\alpha \times \beta\).
‣ DirectProductOnMorphismsWithGivenDirectProducts( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times b, a' \times b')\)
The arguments are an object \(s = a \times b\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = a' \times b'\). The output is the direct product \(\alpha \times \beta\).
‣ CartesianAssociatorRightToLeft( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \times (b \times c), (a \times b) \times c )\).
The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{a,(b,c)}: a \times (b \times c) \rightarrow (a \times b) \times c\).
‣ CartesianAssociatorRightToLeftWithGivenDirectProducts( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \times (b \times c), (a \times b) \times c )\).
The arguments are an object \(s = a \times (b \times c)\), three objects \(a,b,c\), and an object \(r = (a \times b) \times c\). The output is the associator \(\alpha_{a,(b,c)}: a \times (b \times c) \rightarrow (a \times b) \times c\).
‣ CartesianAssociatorLeftToRight( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \times b) \times c \rightarrow a \times (b \times c) )\).
The arguments are three objects \(a,b,c\). The output is the associator \(\alpha_{(a,b),c}: (a \times b) \times c \rightarrow a \times (b \times c)\).
‣ CartesianAssociatorLeftToRightWithGivenDirectProducts( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \times b) \times c \rightarrow a \times (b \times c) )\).
The arguments are an object \(s = (a \times b) \times c\), three objects \(a,b,c\), and an object \(r = a \times (b \times c)\). The output is the associator \(\alpha_{(a,b),c}: (a \times b) \times c \rightarrow a \times (b \times c)\).
‣ CartesianLeftUnitor( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(1 \times a, a)\)
The argument is an object \(a\). The output is the left unitor \(\lambda_a: 1 \times a \rightarrow a\).
‣ CartesianLeftUnitorWithGivenDirectProduct( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(1 \times a, a)\)
The arguments are an object \(a\) and an object \(s = 1 \times a\). The output is the left unitor \(\lambda_a: 1 \times a \rightarrow a\).
‣ CartesianLeftUnitorInverse( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, 1 \times a)\)
The argument is an object \(a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \times a\).
‣ CartesianLeftUnitorInverseWithGivenDirectProduct( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, 1 \times a)\)
The argument is an object \(a\) and an object \(r = 1 \times a\). The output is the inverse of the left unitor \(\lambda_a^{-1}: a \rightarrow 1 \times a\).
‣ CartesianRightUnitor( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a \times 1, a)\)
The argument is an object \(a\). The output is the right unitor \(\rho_a: a \times 1 \rightarrow a\).
‣ CartesianRightUnitorWithGivenDirectProduct( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times 1, a)\)
The arguments are an object \(a\) and an object \(s = a \times 1\). The output is the right unitor \(\rho_a: a \times 1 \rightarrow a\).
‣ CartesianRightUnitorInverse( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, a \times 1)\)
The argument is an object \(a\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \times 1\).
‣ CartesianRightUnitorInverseWithGivenDirectProduct( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, a \times 1)\)
The arguments are an object \(a\) and an object \(r = a \times 1\). The output is the inverse of the right unitor \(\rho_a^{-1}: a \rightarrow a \times 1\).
‣ CartesianDiagonal( a, n ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, a^{\times n})\).
The arguments are an object \(a\) and an integer \(n \geq 0\). The output is the diagonal morphism from \(a\) to the \(n\)-fold cartesian power \(a^{\times n}\). If the category does not support empty limits, \(n\) must be not be 0.
‣ CartesianDiagonalWithGivenCartesianPower( a, n, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, b)\)
The arguments are an object \(a\), an integer \(n\), and an object \(b\) equal to the \(n\)-fold cartesian power \(a^{\times n}\) of \(a\). The output is the diagonal morphism from \(a\) to \(b\).
‣ LeftCartesianDistributivityExpanding( a, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a \times (b_1 \sqcup \dots \sqcup b_n), (a \times b_1) \sqcup \dots \sqcup (a \times b_n) )\)
The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \(a \times (b_1 \sqcup \dots \sqcup b_n) \rightarrow (a \times b_1) \sqcup \dots \sqcup (a \times b_n)\).
‣ LeftCartesianDistributivityExpandingWithGivenObjects( s, a, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = a \times (b_1 \sqcup \dots \sqcup b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = (a \times b_1) \sqcup \dots \sqcup (a \times b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).
‣ LeftCartesianDistributivityFactoring( a, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (a \times b_1) \sqcup \dots \sqcup (a \times b_n), a \times (b_1 \sqcup \dots \sqcup b_n) )\)
The arguments are an object \(a\) and a list of objects \(L = (b_1, \dots, b_n)\). The output is the left distributivity morphism \((a \times b_1) \sqcup \dots \sqcup (a \times b_n) \rightarrow a \times (b_1 \sqcup \dots \sqcup b_n)\).
‣ LeftCartesianDistributivityFactoringWithGivenObjects( s, a, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (a \times b_1) \sqcup \dots \sqcup (a \times b_n)\), an object \(a\), a list of objects \(L = (b_1, \dots, b_n)\), and an object \(r = a \times (b_1 \sqcup \dots \sqcup b_n)\). The output is the left distributivity morphism \(s \rightarrow r\).
‣ RightCartesianDistributivityExpanding( L, a ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (b_1 \sqcup \dots \sqcup b_n) \times a, (b_1 \times a) \sqcup \dots \sqcup (b_n \times a) )\)
The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \sqcup \dots \sqcup b_n) \times a \rightarrow (b_1 \times a) \sqcup \dots \sqcup (b_n \times a)\).
‣ RightCartesianDistributivityExpandingWithGivenObjects( s, L, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (b_1 \sqcup \dots \sqcup b_n) \times a\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \times a) \sqcup \dots \sqcup (b_n \times a)\). The output is the right distributivity morphism \(s \rightarrow r\).
‣ RightCartesianDistributivityFactoring( L, a ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( (b_1 \times a) \sqcup \dots \sqcup (b_n \times a), (b_1 \sqcup \dots \sqcup b_n) \times a)\)
The arguments are a list of objects \(L = (b_1, \dots, b_n)\) and an object \(a\). The output is the right distributivity morphism \((b_1 \times a) \sqcup \dots \sqcup (b_n \times a) \rightarrow (b_1 \sqcup \dots \sqcup b_n) \times a \).
‣ RightCartesianDistributivityFactoringWithGivenObjects( s, L, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = (b_1 \times a) \sqcup \dots \sqcup (b_n \times a)\), a list of objects \(L = (b_1, \dots, b_n)\), an object \(a\), and an object \(r = (b_1 \sqcup \dots \sqcup b_n) \times a\). The output is the right distributivity morphism \(s \rightarrow r\).
A cartesian category \(\mathbf{C}\) which has for each functor \(- \times b: \mathbf{C} \rightarrow \mathbf{C}\) a right adjoint (denoted by \(\mathrm{Exponential}(b,-)\)) is called a closed cartesian category.
The corresponding GAP property is called IsCartesianClosedCategory.
‣ ExponentialOnObjects( a, b ) | ( operation ) |
Returns: an object
The arguments are two objects \(a,b\). The output is the exponential object \(\mathrm{Exponential}(a,b)\).
‣ ExponentialOnMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Exponential}(a',b), \mathrm{Exponential}(a,b') )\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\). The output is the exponential morphism \(\mathrm{Exponential}(\alpha,\beta): \mathrm{Exponential}(a',b) \rightarrow \mathrm{Exponential}(a,b')\).
‣ ExponentialOnMorphismsWithGivenExponentials( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = \mathrm{Exponential}(a',b)\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\), and an object \(r = \mathrm{Exponential}(a,b')\). The output is the exponential morphism \(\mathrm{Exponential}(\alpha,\beta): \mathrm{Exponential}(a',b) \rightarrow \mathrm{Exponential}(a,b')\).
‣ CartesianRightEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times \mathrm{Exponential}(a,b), b )\).
The arguments are two objects \(a, b\). The output is the right evaluation morphism \(\mathrm{ev}_{a,b}:a \times \mathrm{Exponential}(a,b) \rightarrow b\), i.e., the counit of the direct product-exponential adjunction.
‣ CartesianRightEvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, b )\).
The arguments are two objects \(a,b\) and an object \(s = a \times \mathrm{Exponential}(a,b)\). The output is the right evaluation morphism \(\mathrm{ev}_{a,b}: a \times \mathrm{Exponential}(a,b) \rightarrow b\), i.e., the counit of the direct product-exponential adjunction.
‣ CartesianRightCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, \mathrm{Exponential}(a, a \times b) )\).
The arguments are two objects \(a,b\). The output is the right coevaluation morphism \(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, a \times b)\), i.e., the unit of the direct product-exponential adjunction.
‣ CartesianRightCoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, r )\).
The arguments are two objects \(a,b\) and an object \(r = \mathrm{Exponential}(a, a \times b)\). The output is the right coevaluation morphism \(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, a \times b)\), i.e., the unit of the direct product-exponential adjunction.
‣ DirectProductToExponentialRightAdjunctMorphism( a, b, f ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, \mathrm{Exponential}(a,c) )\).
The arguments are two objects \(a,b\) and a morphism \(f: a \times b \rightarrow c\). The output is a morphism \(g: b \rightarrow \mathrm{Exponential}(a,c)\) corresponding to \(f\) under the direct product-exponential adjunction.
‣ DirectProductToExponentialRightAdjunctMorphismWithGivenExponential( a, b, f, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, i )\).
The arguments are two objects \(a,b\), a morphism \(f: a \times b \rightarrow c\) and an object \(i = \mathrm{Exponential}(a,c)\). The output is a morphism \(g: b \rightarrow \mathrm{Exponential}(a,c)\) corresponding to \(f\) under the direct product-exponential adjunction.
‣ ExponentialToDirectProductRightAdjunctMorphism( a, c, g ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times b, c)\).
The arguments are two objects \(a,c\) and a morphism \(g: b \rightarrow \mathrm{Exponential}(a,c)\). The output is a morphism \(f: a \times b \rightarrow c\) corresponding to \(g\) under the direct product-exponential adjunction.
‣ ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct( a, c, g, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, c)\).
The arguments are two objects \(a,c\), a morphism \(g: b \rightarrow \mathrm{Exponential}(a,c)\) and an object \(s = a \times b\). The output is a morphism \(f: s \rightarrow c\) corresponding to \(g\) under the direct product-exponential adjunction.
‣ CartesianLeftEvaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Exponential}(a,b) \times a, b )\).
The arguments are two objects \(a, b\). The output is the left evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{Exponential}(a,b) \times a \rightarrow b\), i.e., the counit of the direct product-exponential adjunction.
‣ CartesianLeftEvaluationMorphismWithGivenSource( a, b, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, b )\).
The arguments are two objects \(a,b\) and an object \(s = \mathrm{Exponential}(a,b) \times a\). The output is the left evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{Exponential}(a,b) \times a \rightarrow b\), i.e., the counit of the direct product-exponential adjunction.
‣ CartesianLeftCoevaluationMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, \mathrm{Exponential}(a, b \times a) )\).
The arguments are two objects \(a,b\). The output is the left coevaluation morphism \(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, b \times a)\), i.e., the unit of the direct product-exponential adjunction.
‣ CartesianLeftCoevaluationMorphismWithGivenRange( a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( b, r )\).
The arguments are two objects \(a,b\) and an object \(r = \mathrm{Exponential}(a, b \times a)\). The output is the left coevaluation morphism \(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, b \times a)\), i.e., the unit of the direct product-exponential adjunction.
‣ DirectProductToExponentialLeftAdjunctMorphism( a, b, f ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, \mathrm{Exponential}(b,c) )\).
The arguments are two objects \(a,b\) and a morphism \(f: a \times b \rightarrow c\). The output is a morphism \(g: a \rightarrow \mathrm{Exponential}(b,c)\) corresponding to \(f\) under the direct product-exponential adjunction.
‣ DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential( a, b, f, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, i )\).
The arguments are two objects \(a,b\), a morphism \(f: a \times b \rightarrow c\) and an object \(i = \mathrm{Exponential}(b,c)\). The output is a morphism \(g: a \rightarrow \mathrm{Exponential}(b,c)\) corresponding to \(f\) under the direct product-exponential adjunction.
‣ ExponentialToDirectProductLeftAdjunctMorphism( b, c, g ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a \times b, c)\).
The arguments are two objects \(b,c\) and a morphism \(g: a \rightarrow \mathrm{Exponential}(b,c)\). The output is a morphism \(f: a \times b \rightarrow c\) corresponding to \(g\) under the direct product-exponential adjunction.
‣ ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct( b, c, g, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, c)\).
The arguments are two objects \(b,c\), a morphism \(g: a \rightarrow \mathrm{Exponential}(b,c)\) and an object \(s = a \times b\). The output is a morphism \(f: s \rightarrow c\) corresponding to \(g\) under the direct product-exponential adjunction.
‣ CartesianPreComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c), \mathrm{Exponential}(a,c) )\).
The arguments are three objects \(a,b,c\). The output is the precomposition morphism \(\mathrm{CartesianPreComposeMorphism}_{a,b,c}: \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c) \rightarrow \mathrm{Exponential}(a,c)\).
‣ CartesianPreComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are an object \(s = \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c)\), three objects \(a,b,c\), and an object \(r = \mathrm{Exponential}(a,c)\). The output is the precomposition morphism \(\mathrm{CartesianPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c) \rightarrow \mathrm{Exponential}(a,c)\).
‣ CartesianPostComposeMorphism( a, b, c ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b), \mathrm{Exponential}(a,c) )\).
The arguments are three objects \(a,b,c\). The output is the postcomposition morphism \(\mathrm{CartesianPostComposeMorphism}_{a,b,c}: \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b) \rightarrow \mathrm{Exponential}(a,c)\).
‣ CartesianPostComposeMorphismWithGivenObjects( s, a, b, c, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are an object \(s = \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b)\), three objects \(a,b,c\), and an object \(r = \mathrm{Exponential}(a,c)\). The output is the postcomposition morphism \(\mathrm{CartesianPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b) \rightarrow \mathrm{Exponential}(a,c)\).
‣ CartesianDualOnObjects( a ) | ( attribute ) |
Returns: an object
The argument is an object \(a\). The output is its dual object \(a^{\vee}\).
‣ CartesianDualOnMorphisms( 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 \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).
‣ CartesianDualOnMorphismsWithGivenCartesianDuals( s, alpha, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The argument is an object \(s = b^{\vee}\), a morphism \(\alpha: a \rightarrow b\), and an object \(r = a^{\vee}\). The output is the dual morphism \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).
‣ CartesianEvaluationForCartesianDual( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \times a, 1 )\).
The argument is an object \(a\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \times a \rightarrow 1\).
‣ CartesianEvaluationForCartesianDualWithGivenDirectProduct( s, a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are an object \(s = a^{\vee} \times a\), an object \(a\), and an object \(r = 1\). The output is the evaluation morphism \(\mathrm{ev}_{a}: a^{\vee} \times a \rightarrow 1\).
‣ MorphismToCartesianBidual( 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 \(a \rightarrow (a^{\vee})^{\vee}\).
‣ MorphismToCartesianBidualWithGivenCartesianBidual( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, r)\).
The arguments are an object \(a\), and an object \(r = (a^{\vee})^{\vee}\). The output is the morphism to the bidual \(a \rightarrow (a^{\vee})^{\vee}\).
‣ DirectProductExponentialCompatibilityMorphism( list ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b'), \mathrm{Exponential}(a \times b,a' \times b'))\).
The argument is a list of four objects \([ a, a', b, b' ]\). The output is the natural morphism \(\mathrm{DirectProductExponentialCompatibilityMorphism}_{a,a',b,b'}: \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b') \rightarrow \mathrm{Exponential}(a \times b,a' \times b')\).
‣ DirectProductExponentialCompatibilityMorphismWithGivenObjects( s, list, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are a list of four objects \([ a, a', b, b' ]\), and two objects \(s = \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b')\) and \(r = \mathrm{Exponential}(a \times b,a' \times b')\). The output is the natural morphism \(\mathrm{DirectProductExponentialCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b') \rightarrow \mathrm{Exponential}(a \times b,a' \times b')\).
‣ DirectProductCartesianDualityCompatibilityMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \times b^{\vee}, (a \times b)^{\vee} )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{DirectProductCartesianDualityCompatibilityMorphism}: a^{\vee} \times b^{\vee} \rightarrow (a \times b)^{\vee}\).
‣ DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are an object \(s = a^{\vee} \times b^{\vee}\), two objects \(a,b\), and an object \(r = (a \times b)^{\vee}\). The output is the natural morphism \(\mathrm{DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \times b^{\vee} \rightarrow (a \times b)^{\vee}\).
‣ MorphismFromDirectProductToExponential( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a^{\vee} \times b, \mathrm{Exponential}(a,b) )\).
The arguments are two objects \(a,b\). The output is the natural morphism \(\mathrm{MorphismFromDirectProductToExponential}_{a,b}: a^{\vee} \times b \rightarrow \mathrm{Exponential}(a,b)\).
‣ MorphismFromDirectProductToExponentialWithGivenObjects( s, a, b, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\).
The arguments are an object \(s = a^{\vee} \times b\), two objects \(a,b\), and an object \(r = \mathrm{Exponential}(a,b)\). The output is the natural morphism \(\mathrm{MorphismFromDirectProductToExponentialWithGivenObjects}_{a,b}: a^{\vee} \times b \rightarrow \mathrm{Exponential}(a,b)\).
‣ IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a^{\vee}, \mathrm{Exponential}(a,1))\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject}_{a}: a^{\vee} \rightarrow \mathrm{Exponential}(a,1)\).
‣ IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{Exponential}(a,1), a^{\vee})\).
The argument is an object \(a\). The output is the isomorphism \(\mathrm{IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject}_{a}: \mathrm{Exponential}(a,1) \rightarrow a^{\vee}\).
‣ UniversalPropertyOfCartesianDual( t, a, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(t, a^{\vee})\).
The arguments are two objects \(t,a\), and a morphism \(\alpha: t \times a \rightarrow 1\). The output is the morphism \(t \rightarrow a^{\vee}\) given by the universal property of \(a^{\vee}\).
‣ CartesianLambdaIntroduction( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( 1, \mathrm{Exponential}(a,b) )\).
The argument is a morphism \(\alpha: a \rightarrow b\). The output is the corresponding morphism \(1 \rightarrow \mathrm{Exponential}(a,b)\) under the direct product-exponential adjunction.
‣ CartesianLambdaElimination( a, b, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\).
The arguments are two objects \(a,b\), and a morphism \(\alpha: 1 \rightarrow \mathrm{Exponential}(a,b)\). The output is a morphism \(a \rightarrow b\) corresponding to \(\alpha\) under the direct product-exponential adjunction.
‣ IsomorphismFromObjectToExponential( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a, \mathrm{Exponential}(1,a))\).
The argument is an object \(a\). The output is the natural isomorphism \(a \rightarrow \mathrm{Exponential}(1,a)\).
‣ IsomorphismFromObjectToExponentialWithGivenExponential( a, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a, r)\).
The argument is an object \(a\), and an object \(r = \mathrm{Exponential}(1,a)\). The output is the natural isomorphism \(a \rightarrow \mathrm{Exponential}(1,a)\).
‣ IsomorphismFromExponentialToObject( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{Exponential}(1,a),a)\).
The argument is an object \(a\). The output is the natural isomorphism \(\mathrm{Exponential}(1,a) \rightarrow a\).
‣ IsomorphismFromExponentialToObjectWithGivenExponential( a, s ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s,a)\).
The argument is an object \(a\), and an object \(s = \mathrm{Exponential}(1,a)\). The output is the natural isomorphism \(\mathrm{Exponential}(1,a) \rightarrow a\).
‣ Exponential( a, b ) | ( operation ) |
Returns: a cell
This is a convenience method. The arguments are two cells \(a,b\). The output is the exponential cell. If \(a,b\) are two CAP objects the output is the internal Hom object \(\mathrm{Exponential}(a,b)\). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the exponential on morphisms, where any object is replaced by its identity morphism.
‣ AddCartesianBraiding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianBraiding. \(F: ( a, b ) \mapsto \mathtt{CartesianBraiding}(a, b)\).
‣ AddCartesianBraidingInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianBraidingInverse. \(F: ( a, b ) \mapsto \mathtt{CartesianBraidingInverse}(a, b)\).
‣ AddCartesianBraidingInverseWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianBraidingInverseWithGivenDirectProducts. \(F: ( s, a, b, r ) \mapsto \mathtt{CartesianBraidingInverseWithGivenDirectProducts}(s, a, b, r)\).
‣ AddCartesianBraidingWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianBraidingWithGivenDirectProducts. \(F: ( s, a, b, r ) \mapsto \mathtt{CartesianBraidingWithGivenDirectProducts}(s, a, b, r)\).
‣ AddCartesianAssociatorLeftToRight( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianAssociatorLeftToRight. \(F: ( a, b, c ) \mapsto \mathtt{CartesianAssociatorLeftToRight}(a, b, c)\).
‣ AddCartesianAssociatorLeftToRightWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianAssociatorLeftToRightWithGivenDirectProducts. \(F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianAssociatorLeftToRightWithGivenDirectProducts}(s, a, b, c, r)\).
‣ AddCartesianAssociatorRightToLeft( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianAssociatorRightToLeft. \(F: ( a, b, c ) \mapsto \mathtt{CartesianAssociatorRightToLeft}(a, b, c)\).
‣ AddCartesianAssociatorRightToLeftWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianAssociatorRightToLeftWithGivenDirectProducts. \(F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianAssociatorRightToLeftWithGivenDirectProducts}(s, a, b, c, r)\).
‣ AddCartesianDiagonal( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianDiagonal. \(F: ( a, n ) \mapsto \mathtt{CartesianDiagonal}(a, n)\).
‣ AddCartesianDiagonalWithGivenCartesianPower( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianDiagonalWithGivenCartesianPower. \(F: ( a, n, cartesian_power ) \mapsto \mathtt{CartesianDiagonalWithGivenCartesianPower}(a, n, cartesian_power)\).
‣ AddCartesianLeftUnitor( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftUnitor. \(F: ( a ) \mapsto \mathtt{CartesianLeftUnitor}(a)\).
‣ AddCartesianLeftUnitorInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftUnitorInverse. \(F: ( a ) \mapsto \mathtt{CartesianLeftUnitorInverse}(a)\).
‣ AddCartesianLeftUnitorInverseWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftUnitorInverseWithGivenDirectProduct. \(F: ( a, r ) \mapsto \mathtt{CartesianLeftUnitorInverseWithGivenDirectProduct}(a, r)\).
‣ AddCartesianLeftUnitorWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftUnitorWithGivenDirectProduct. \(F: ( a, s ) \mapsto \mathtt{CartesianLeftUnitorWithGivenDirectProduct}(a, s)\).
‣ AddCartesianRightUnitor( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightUnitor. \(F: ( a ) \mapsto \mathtt{CartesianRightUnitor}(a)\).
‣ AddCartesianRightUnitorInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightUnitorInverse. \(F: ( a ) \mapsto \mathtt{CartesianRightUnitorInverse}(a)\).
‣ AddCartesianRightUnitorInverseWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightUnitorInverseWithGivenDirectProduct. \(F: ( a, r ) \mapsto \mathtt{CartesianRightUnitorInverseWithGivenDirectProduct}(a, r)\).
‣ AddCartesianRightUnitorWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightUnitorWithGivenDirectProduct. \(F: ( a, s ) \mapsto \mathtt{CartesianRightUnitorWithGivenDirectProduct}(a, s)\).
‣ AddDirectProductOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductOnMorphisms. \(F: ( alpha, beta ) \mapsto \mathtt{DirectProductOnMorphisms}(alpha, beta)\).
‣ AddDirectProductOnMorphismsWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductOnMorphismsWithGivenDirectProducts. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{DirectProductOnMorphismsWithGivenDirectProducts}(s, alpha, beta, r)\).
‣ AddCartesianDualOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianDualOnMorphisms. \(F: ( alpha ) \mapsto \mathtt{CartesianDualOnMorphisms}(alpha)\).
‣ AddCartesianDualOnMorphismsWithGivenCartesianDuals( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianDualOnMorphismsWithGivenCartesianDuals. \(F: ( s, alpha, r ) \mapsto \mathtt{CartesianDualOnMorphismsWithGivenCartesianDuals}(s, alpha, r)\).
‣ AddCartesianDualOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianDualOnObjects. \(F: ( a ) \mapsto \mathtt{CartesianDualOnObjects}(a)\).
‣ AddCartesianEvaluationForCartesianDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianEvaluationForCartesianDual. \(F: ( a ) \mapsto \mathtt{CartesianEvaluationForCartesianDual}(a)\).
‣ AddCartesianEvaluationForCartesianDualWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianEvaluationForCartesianDualWithGivenDirectProduct. \(F: ( s, a, r ) \mapsto \mathtt{CartesianEvaluationForCartesianDualWithGivenDirectProduct}(s, a, r)\).
‣ AddCartesianLambdaElimination( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLambdaElimination. \(F: ( a, b, alpha ) \mapsto \mathtt{CartesianLambdaElimination}(a, b, alpha)\).
‣ AddCartesianLambdaIntroduction( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLambdaIntroduction. \(F: ( alpha ) \mapsto \mathtt{CartesianLambdaIntroduction}(alpha)\).
‣ AddCartesianLeftCoevaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftCoevaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CartesianLeftCoevaluationMorphism}(a, b)\).
‣ AddCartesianLeftCoevaluationMorphismWithGivenRange( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftCoevaluationMorphismWithGivenRange. \(F: ( a, b, r ) \mapsto \mathtt{CartesianLeftCoevaluationMorphismWithGivenRange}(a, b, r)\).
‣ AddCartesianLeftEvaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftEvaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CartesianLeftEvaluationMorphism}(a, b)\).
‣ AddCartesianLeftEvaluationMorphismWithGivenSource( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianLeftEvaluationMorphismWithGivenSource. \(F: ( a, b, s ) \mapsto \mathtt{CartesianLeftEvaluationMorphismWithGivenSource}(a, b, s)\).
‣ AddCartesianPostComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianPostComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{CartesianPostComposeMorphism}(a, b, c)\).
‣ AddCartesianPostComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianPostComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianPostComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddCartesianPreComposeMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianPreComposeMorphism. \(F: ( a, b, c ) \mapsto \mathtt{CartesianPreComposeMorphism}(a, b, c)\).
‣ AddCartesianPreComposeMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianPreComposeMorphismWithGivenObjects. \(F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianPreComposeMorphismWithGivenObjects}(s, a, b, c, r)\).
‣ AddCartesianRightCoevaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightCoevaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CartesianRightCoevaluationMorphism}(a, b)\).
‣ AddCartesianRightCoevaluationMorphismWithGivenRange( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightCoevaluationMorphismWithGivenRange. \(F: ( a, b, r ) \mapsto \mathtt{CartesianRightCoevaluationMorphismWithGivenRange}(a, b, r)\).
‣ AddCartesianRightEvaluationMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightEvaluationMorphism. \(F: ( a, b ) \mapsto \mathtt{CartesianRightEvaluationMorphism}(a, b)\).
‣ AddCartesianRightEvaluationMorphismWithGivenSource( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CartesianRightEvaluationMorphismWithGivenSource. \(F: ( a, b, s ) \mapsto \mathtt{CartesianRightEvaluationMorphismWithGivenSource}(a, b, s)\).
‣ AddDirectProductCartesianDualityCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductCartesianDualityCompatibilityMorphism. \(F: ( a, b ) \mapsto \mathtt{DirectProductCartesianDualityCompatibilityMorphism}(a, b)\).
‣ AddDirectProductCartesianDualityCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r)\).
‣ AddDirectProductExponentialCompatibilityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductExponentialCompatibilityMorphism. \(F: ( list ) \mapsto \mathtt{DirectProductExponentialCompatibilityMorphism}(list)\).
‣ AddDirectProductExponentialCompatibilityMorphismWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductExponentialCompatibilityMorphismWithGivenObjects. \(F: ( source, list, range ) \mapsto \mathtt{DirectProductExponentialCompatibilityMorphismWithGivenObjects}(source, list, range)\).
‣ AddDirectProductToExponentialLeftAdjunctMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductToExponentialLeftAdjunctMorphism. \(F: ( a, b, f ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctMorphism}(a, b, f)\).
‣ AddDirectProductToExponentialLeftAdjunctMorphismWithGivenExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential. \(F: ( a, b, f, i ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential}(a, b, f, i)\).
‣ AddDirectProductToExponentialRightAdjunctMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductToExponentialRightAdjunctMorphism. \(F: ( a, b, f ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctMorphism}(a, b, f)\).
‣ AddDirectProductToExponentialRightAdjunctMorphismWithGivenExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductToExponentialRightAdjunctMorphismWithGivenExponential. \(F: ( a, b, f, i ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctMorphismWithGivenExponential}(a, b, f, i)\).
‣ AddExponentialOnMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialOnMorphisms. \(F: ( alpha, beta ) \mapsto \mathtt{ExponentialOnMorphisms}(alpha, beta)\).
‣ AddExponentialOnMorphismsWithGivenExponentials( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialOnMorphismsWithGivenExponentials. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{ExponentialOnMorphismsWithGivenExponentials}(s, alpha, beta, r)\).
‣ AddExponentialOnObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialOnObjects. \(F: ( a, b ) \mapsto \mathtt{ExponentialOnObjects}(a, b)\).
‣ AddExponentialToDirectProductLeftAdjunctMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialToDirectProductLeftAdjunctMorphism. \(F: ( b, c, g ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctMorphism}(b, c, g)\).
‣ AddExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct. \(F: ( b, c, g, s ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct}(b, c, g, s)\).
‣ AddExponentialToDirectProductRightAdjunctMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialToDirectProductRightAdjunctMorphism. \(F: ( a, c, g ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctMorphism}(a, c, g)\).
‣ AddExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct. \(F: ( a, c, g, s ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct}(a, c, g, s)\).
‣ AddIsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject}(a)\).
‣ AddIsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject}(a)\).
‣ AddIsomorphismFromExponentialToObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromExponentialToObject. \(F: ( a ) \mapsto \mathtt{IsomorphismFromExponentialToObject}(a)\).
‣ AddIsomorphismFromExponentialToObjectWithGivenExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromExponentialToObjectWithGivenExponential. \(F: ( a, s ) \mapsto \mathtt{IsomorphismFromExponentialToObjectWithGivenExponential}(a, s)\).
‣ AddIsomorphismFromObjectToExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToExponential. \(F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToExponential}(a)\).
‣ AddIsomorphismFromObjectToExponentialWithGivenExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromObjectToExponentialWithGivenExponential. \(F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToExponentialWithGivenExponential}(a, r)\).
‣ AddMorphismFromDirectProductToExponential( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromDirectProductToExponential. \(F: ( a, b ) \mapsto \mathtt{MorphismFromDirectProductToExponential}(a, b)\).
‣ AddMorphismFromDirectProductToExponentialWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromDirectProductToExponentialWithGivenObjects. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromDirectProductToExponentialWithGivenObjects}(s, a, b, r)\).
‣ AddMorphismToCartesianBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToCartesianBidual. \(F: ( a ) \mapsto \mathtt{MorphismToCartesianBidual}(a)\).
‣ AddMorphismToCartesianBidualWithGivenCartesianBidual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismToCartesianBidualWithGivenCartesianBidual. \(F: ( a, r ) \mapsto \mathtt{MorphismToCartesianBidualWithGivenCartesianBidual}(a, r)\).
‣ AddUniversalPropertyOfCartesianDual( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalPropertyOfCartesianDual. \(F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCartesianDual}(t, a, alpha)\).
‣ AddLeftCartesianDistributivityExpanding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftCartesianDistributivityExpanding. \(F: ( a, L ) \mapsto \mathtt{LeftCartesianDistributivityExpanding}(a, L)\).
‣ AddLeftCartesianDistributivityExpandingWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftCartesianDistributivityExpandingWithGivenObjects. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftCartesianDistributivityExpandingWithGivenObjects}(s, a, L, r)\).
‣ AddLeftCartesianDistributivityFactoring( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftCartesianDistributivityFactoring. \(F: ( a, L ) \mapsto \mathtt{LeftCartesianDistributivityFactoring}(a, L)\).
‣ AddLeftCartesianDistributivityFactoringWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation LeftCartesianDistributivityFactoringWithGivenObjects. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftCartesianDistributivityFactoringWithGivenObjects}(s, a, L, r)\).
‣ AddRightCartesianDistributivityExpanding( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightCartesianDistributivityExpanding. \(F: ( L, a ) \mapsto \mathtt{RightCartesianDistributivityExpanding}(L, a)\).
‣ AddRightCartesianDistributivityExpandingWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightCartesianDistributivityExpandingWithGivenObjects. \(F: ( s, L, a, r ) \mapsto \mathtt{RightCartesianDistributivityExpandingWithGivenObjects}(s, L, a, r)\).
‣ AddRightCartesianDistributivityFactoring( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightCartesianDistributivityFactoring. \(F: ( L, a ) \mapsto \mathtt{RightCartesianDistributivityFactoring}(L, a)\).
‣ AddRightCartesianDistributivityFactoringWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RightCartesianDistributivityFactoringWithGivenObjects. \(F: ( s, L, a, r ) \mapsto \mathtt{RightCartesianDistributivityFactoringWithGivenObjects}(s, L, a, r)\).
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