‣ MorphismBetweenCoproducts( diagram_S, F, diagram_T ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\sqcup_{i=0}^{m-1}A_i, \sqcup_{j=0}^{n-1} B_j)\)
The arguments are given as follows:
diagram_S is a list of objects \((A_i)_{i = 0 \dots m - 1}\),
diagram_T is a list of objects \((B_j)_{j = 0 \dots n - 1}\),
F is a pair consisting of a list \(f = (f_i)_{i = 0 \dots m - 1}\) of length \(m\) of integers between \(0, \ldots, n - 1\) and a list of morphisms \(( A_i \rightarrow B_{f_i} )_{i = 0 \dots m - 1}\).
The output is the morphism \(\sqcup_{i=0}^{m-1} A_i \rightarrow \sqcup_{j=0}^{n-1} B_j\) defined by the pair F.
gap> LoadPackage( "ToolsForCategoricalTowers" ); true gap> LoadPackage( "Toposes", ">= 2024.02-08", false ); true gap> LoadPackage( "FinSetsForCAP", ">= 2024.02-09", false ); true gap> sFinSets := SkeletalCategoryOfFiniteSets( ); SkeletalFinSets gap> Display( sFinSets ); A CAP category with name SkeletalFinSets: 58 primitive operations were used to derive 342 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsElementaryTopos and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> a := FinSet( sFinSets, 2 ); |2| gap> b := FinSet( sFinSets, 3 ); |3| gap> beta := CocartesianBraiding( a, b ); |5| → |5| gap> Display( beta ); { 0,..., 4 } ⱶ[ 3, 4, 0, 1, 2 ]→ { 0,..., 4 } gap> id_a := IdentityMorphism( a ); |2| → |2| gap> id_b := IdentityMorphism( b ); |3| → |3| gap> f := Pair( [ 1, 0 ], [ id_a, id_b ] ); [ [ 1, 0 ], [ |2| → |2|, |3| → |3| ] ] gap> beta2 := MorphismBetweenCoproducts( [ a, b ], f, [ b, a ] ); |5| → |5| gap> beta2 = beta; true gap> W := WrapperCategory( sFinSets, > rec( only_primitive_operations := false ) ); WrapperCategory( SkeletalFinSets ) gap> Display( W ); A CAP category with name WrapperCategory( SkeletalFinSets ): 326 primitive operations were used to derive 340 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsElementaryTopos and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> W_a := a / W; <An object in WrapperCategory( SkeletalFinSets )> gap> W_b := b / W; <An object in WrapperCategory( SkeletalFinSets )> gap> W_beta := CocartesianBraiding( W_a, W_b ); <A morphism in WrapperCategory( SkeletalFinSets )> gap> MorphismDatum( W_beta ) = beta; true gap> W_id_a := IdentityMorphism( W_a ); <An identity morphism in WrapperCategory( SkeletalFinSets )> gap> W_id_b := IdentityMorphism( W_b ); <An identity morphism in WrapperCategory( SkeletalFinSets )> gap> W_f := Pair( [ 1, 0 ], [ W_id_a, W_id_b ] );; gap> W_beta2 := MorphismBetweenCoproducts( [ W_a, W_b ], W_f, [ W_b, W_a ] ); <A morphism in WrapperCategory( SkeletalFinSets )> gap> W_beta2 = W_beta; true gap> O := Opposite( sFinSets ); Opposite( SkeletalFinSets ) gap> Display( O ); A CAP category with name Opposite( SkeletalFinSets ): 259 primitive operations were used to derive 275 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsBicartesianCoclosedCategory * IsFiniteBicompleteCategory and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> O_a := a / O; <An object in Opposite( SkeletalFinSets )> gap> O_b := b / O; <An object in Opposite( SkeletalFinSets )> gap> O_beta := CartesianBraiding( O_b, O_a ); <A morphism in Opposite( SkeletalFinSets )> gap> MorphismDatum( O_beta ) = beta; true gap> O_id_a := IdentityMorphism( O_a ); <An identity morphism in Opposite( SkeletalFinSets )> gap> O_id_b := IdentityMorphism( O_b ); <An identity morphism in Opposite( SkeletalFinSets )> gap> O_f := Pair( [ 1, 0 ], [ O_id_a, O_id_b ] );; gap> O_beta2 := MorphismBetweenDirectProducts( [ O_b, O_a ], O_f, [ O_a, O_b ] ); <A morphism in Opposite( SkeletalFinSets )> gap> O_beta2 = O_beta; true
‣ MorphismBetweenCoproductsWithGivenCoproducts( S, diagram_S, F, diagram_T, T ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\sqcup_{i=0}^{m-1}A_i, \sqcup_{j=0}^{n-1} B_j)\)
The arguments are given as follows:
diagram_S is a list of objects \((A_i)_{i = 0 \dots m - 1}\),
diagram_T is a list of objects \((B_j)_{j = 0 \dots n - 1}\),
S is the coproduct \(\sqcup_{i=0}^{m-1} A_i\),
T is the coproduct \(\sqcup_{j=0}^{n-1} B_j\),
F is a pair consisting of a list \(f = (f_i)_{i = 0 \dots m - 1}\) of length \(m\) of integers between \(0, \ldots, n - 1\) and a list of morphisms \(( A_i \rightarrow B_{f_i} )_{i = 0 \dots m - 1}\).
The output is the morphism \(\sqcup_{i=0}^{m-1} A_i \rightarrow \sqcup_{j=0}^{n-1} B_j\) defined by the pair F.
‣ MorphismBetweenDirectProducts( diagram_S, F, diagram_T ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\times_{i=0}^{m-1}A_i, \times_{j=0}^{n-1} B_j)\)
The arguments are given as follows:
diagram_S is a list of objects \((A_i)_{i = 0 \dots m - 1}\),
diagram_T is a list of objects \((B_j)_{j = 0 \dots n - 1}\),
F is a pair consisting of a list \(f = (f_j)_{j = 0 \dots n - 1}\) of length \(m\) of integers between \(0, \ldots, m - 1\) and a list of morphisms \(( B_j \rightarrow A_{f_j} )_{j = 0 \dots n - 1}\).
The output is the morphism \(\times_{i=0}^{m-1} A_i \rightarrow \times_{j=0}^{n-1} B_j\) defined by the pair F.
gap> LoadPackage( "ToolsForCategoricalTowers" ); true gap> LoadPackage( "Toposes", ">= 2024.02-08", false ); true gap> LoadPackage( "FinSetsForCAP", ">= 2024.02-09", false ); true gap> sFinSets := SkeletalCategoryOfFiniteSets( ); SkeletalFinSets gap> Display( sFinSets ); A CAP category with name SkeletalFinSets: 58 primitive operations were used to derive 342 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsElementaryTopos and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> a := FinSet( sFinSets, 2 ); |2| gap> b := FinSet( sFinSets, 3 ); |3| gap> beta := CartesianBraiding( a, b ); |6| → |6| gap> Display( beta ); { 0,..., 5 } ⱶ[ 0, 3, 1, 4, 2, 5 ]→ { 0,..., 5 } gap> id_a := IdentityMorphism( a ); |2| → |2| gap> id_b := IdentityMorphism( b ); |3| → |3| gap> f := Pair( [ 1, 0 ], [ id_b, id_a ] ); [ [ 1, 0 ], [ |3| → |3|, |2| → |2| ] ] gap> beta2 := MorphismBetweenDirectProducts( [ a, b ], f, [ b, a ] ); |6| → |6| gap> beta2 = beta; true gap> W := WrapperCategory( sFinSets, > rec( only_primitive_operations := false ) ); WrapperCategory( SkeletalFinSets ) gap> Display( W ); A CAP category with name WrapperCategory( SkeletalFinSets ): 326 primitive operations were used to derive 340 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsElementaryTopos and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> W_a := a / W; <An object in WrapperCategory( SkeletalFinSets )> gap> W_b := b / W; <An object in WrapperCategory( SkeletalFinSets )> gap> W_beta := CartesianBraiding( W_a, W_b ); <A morphism in WrapperCategory( SkeletalFinSets )> gap> MorphismDatum( W_beta ) = beta; true gap> W_id_a := IdentityMorphism( W_a ); <An identity morphism in WrapperCategory( SkeletalFinSets )> gap> W_id_b := IdentityMorphism( W_b ); <An identity morphism in WrapperCategory( SkeletalFinSets )> gap> W_f := Pair( [ 1, 0 ], [ W_id_b, W_id_a ] );; gap> W_beta2 := MorphismBetweenDirectProducts( [ W_a, W_b ], W_f, [ W_b, W_a ] ); <A morphism in WrapperCategory( SkeletalFinSets )> gap> W_beta2 = W_beta; true gap> O := Opposite( sFinSets ); Opposite( SkeletalFinSets ) gap> Display( O ); A CAP category with name Opposite( SkeletalFinSets ): 259 primitive operations were used to derive 275 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsBicartesianCoclosedCategory * IsFiniteBicompleteCategory and furthermore mathematically * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> O_a := a / O; <An object in Opposite( SkeletalFinSets )> gap> O_b := b / O; <An object in Opposite( SkeletalFinSets )> gap> O_beta := CocartesianBraiding( O_b, O_a ); <A morphism in Opposite( SkeletalFinSets )> gap> MorphismDatum( O_beta ) = beta; true gap> O_id_a := IdentityMorphism( O_a ); <An identity morphism in Opposite( SkeletalFinSets )> gap> O_id_b := IdentityMorphism( O_b ); <An identity morphism in Opposite( SkeletalFinSets )> gap> O_f := Pair( [ 1, 0 ], [ O_id_b, O_id_a ] );; gap> O_beta2 := MorphismBetweenCoproducts( [ O_b, O_a ], O_f, [ O_a, O_b ] ); <A morphism in Opposite( SkeletalFinSets )> gap> O_beta2 = O_beta; true
‣ MorphismBetweenDirectProductsWithGivenDirectProducts( S, diagram_S, F, diagram_T, T ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\times_{i=0}^{m-1}A_i, \times_{j=0}^{n-1} B_j)\)
The arguments are given as follows:
diagram_S is a list of objects \((A_i)_{i = 0 \dots m - 1}\),
diagram_T is a list of objects \((B_j)_{j = 0 \dots n - 1}\),
S is the product \(\times_{i=0}^{m-1} A_i\),
T is the product \(\times_{j=0}^{n-1} B_j\),
F is a pair consisting of a list \(f = (f_j)_{j = 0 \dots n - 1}\) of length \(m\) of integers between \(0, \ldots, m - 1\) and a list of morphisms \(( B_j \rightarrow A_{f_j} )_{j = 0 \dots n - 1}\).
The output is the morphism \(\times_{i=0}^{m-1} A_i \rightarrow \times_{j=0}^{n-1} B_j\) defined by the pair F.
‣ IsWeakTerminal( a ) | ( property ) |
Returns: a boolean
The argument is an object \(a\) of a category \(\mathbf{C}\). The output is true if \(a\) is weak terminal \(\mathbf{C}\), otherwise the output is false.
‣ IsWeakInitial( a ) | ( property ) |
Returns: a boolean
The argument is an object \(a\) of a category \(\mathbf{C}\). The output is true if \(a\) is weak initial in \(\mathbf{C}\), otherwise the output is false.
‣ RelativeLift( beta, alpha, nu ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(b,a) + \{ \mathtt{fail} \}\)
The arguments are two morphisms \(\beta: b \rightarrow c\), \(\alpha: a \rightarrow c\), and \(\nu: d \rightarrow c\) . The output is a relative lift \(\beta / \alpha: b \rightarrow a\) of \(\beta\) along \(\alpha\) modulo \(\nu\). if such a relative lift exists or \(\mathtt{fail}\) if it doesn't. Recall that a relative lift \(\beta / \alpha: b \rightarrow a\) of \(\beta\) along \(\alpha\) modulo \(\nu\) is a morphism such that \(\alpha \circ (\beta / \alpha) \sim_{b,c} \beta + \chi \nu\), for some morphism \(\chi: b \rightarrow d\).
For a given pair of morphisms \((\alpha: A \rightarrow B, \beta \colon C \rightarrow B)\), a biased relative weak fiber product of \((\alpha, \beta)\) consists of three parts:
an object \(P\),
a morphism \(\pi: P \rightarrow A\) such that there exists a morphism \(\delta: P \rightarrow C\) such that \(\beta \circ \delta \sim_{P,B} \alpha \circ \pi\),
a dependent function \(u\) mapping each \(\tau: T \rightarrow A\), which admits a morphism \(\mu \colon T \rightarrow C\) with \(\beta \circ \mu \sim_{T,B} \alpha \circ \tau\), to a morphism \(u(\tau):T \rightarrow P\) such that \(\pi \circ u(\tau) \sim_{T,A} \tau\).
The triple \(( P, \pi, u )\) is called a biased relative weak fiber product of \((\alpha,\beta)\). We denote the object \(P\) of such a triple by \(\mathrm{BiasedRelativeWeakFiberProduct}(\alpha,\beta)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the biased relative weak fiber product.
‣ BiasedRelativeWeakFiberProduct( alpha, beta, sigma ) | ( operation ) |
Returns: an object
The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the biased relative weak fiber product \(P\) of \(\alpha\) and \(\beta\).
‣ ProjectionOfBiasedRelativeWeakFiberProduct( alpha, beta, sigma ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, A )\)
The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\sigma: D \rightarrow A\). The output is the biased relative weak fiber product projection \(\pi: P \rightarrow A\).
‣ ProjectionOfBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct( alpha, beta, sigma, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, A )\)
The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\sigma: D \rightarrow A\), and an object \(P = \mathrm{BiasedRelativeWeakFiberProduct}( \alpha, \beta )\). The output is the biased relative weak fiber product projection \(\pi: P \rightarrow A\).
‣ UniversalMorphismIntoBiasedRelativeWeakFiberProduct( alpha, beta, sigma, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, P )\)
The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\sigma: D \rightarrow A\), \(\tau: T \rightarrow A\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased relative weak fiber product \(P\) of \(\alpha\) and \(\beta\).
‣ UniversalMorphismIntoBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct( alpha, beta, sigma, tau, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, P )\)
The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\tau: T \rightarrow A\) and an object \(P = \mathrm{BiasedRelativeWeakFiberProduct}( \alpha, \beta )\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased relative weak fiber product \(P\) of \(\alpha\) and \(\beta\).
‣ MorphismOntoSumOfImagesOfAllMorphisms( a, b ) | ( operation ) |
Returns: a morphism
Return a morphism onto the sum of images of all morphisms between a and b.
‣ EmbeddingOfSumOfImagesOfAllMorphisms( a, b ) | ( operation ) |
Returns: a morphism
Return the embedding of the sum of images of all morphisms between a and b.
‣ SumOfImagesOfAllMorphisms( a, b ) | ( operation ) |
Returns: a morphism
Return the sum of images of all morphisms between a and b.
‣ MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory( left_coeffs, right_coeffs, right_side ) | ( operation ) |
Returns: a boolean
Like SolveLinearSystemInAbCategory, but the output is simply true if a unique solution exists, and false otherwise.
‣ MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory( left_coeffs, right_coeffs ) | ( operation ) |
Returns: a boolean
Like BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory, but the output is true if the homogeneous system has only the trivial solution, and false otherwise.
‣ BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory( alpha, beta ) | ( operation ) |
Returns: a list of lists of morphisms \([X^1, \dots, X^t]\)
The arguments are two lists of lists \(\alpha\) and \(\beta\) of morphisms in a linear category over a commutative ring \(k\). The first list \(\alpha\) (the left coefficients) is a list of list of morphisms \(\alpha_{ij}: A_i \rightarrow B_j\), where \(i = 1, \dots, m\) and \(j = 1, \dots, n\) for integers \(m,n \geq 1\). The second list \(\beta\) (the right coefficients) is a list of list of morphisms \(\beta_{ij}: C_j \rightarrow D_i\), where \(i = 1, \dots, m\) and \(j = 1, \dots, n\). The output is a generating set \([X^1,\dots,X^t]\) for the solutions of the homogeneous linear system: \(\sum_{j = 1}^n \alpha_{ij}\cdot X_{j} \cdot \beta_{ij} = 0, ~i = 1, \dots, m\). Particularly, each \(X^k\) is a list (of length \(n\)) of morphisms \(X^k_j:B_j \to C_j, j=1,\dots,n\).
‣ BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory( alpha, beta, gamma, delta ) | ( operation ) |
Returns: a list of lists of morphisms \([X^1, \dots, X^t]\)
The arguments are four lists of lists \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) of morphisms in some linear category over commutative ring. Each of \(\alpha\) and \(\gamma\) is a list of list of morphisms \(\alpha_{ij}, \gamma_{ij}: A_i \rightarrow B_j\), where \(i = 1, \dots, m\) and \(j = 1, \dots, n\) for integers \(m,n \geq 1\). Each of \(\beta\) and \(\delta\) is also a list of list of morphisms \(\beta_{ij}, \delta_{ij}: C_j \rightarrow D_i\), where \(i = 1, \dots, m\) and \(j = 1, \dots, n\). The output is a generating set \([X^1,\dots,X^t]\) for the solutions of the homogeneous linear system defined by \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\), i.e., \(\sum_{j = 1}^n \alpha_{ij}\cdot X^{k}_{j} \cdot \beta_{ij} = \sum_{j = 1}^n \gamma_{ij}\cdot X^{k}_{j} \cdot \delta_{ij}\) for all \(i = 1, \dots, m\) and all \(k = 1, \dots, t\).
‣ BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory( alpha, delta ) | ( operation ) |
Returns: a list of lists of morphisms \([X^1, \dots, X^t]\)
The arguments are two lists of lists \(\alpha\), \(\delta\) morphisms in some linear category over commutative ring. \(\alpha\) is a list of list of morphisms \(\alpha_{ij}:A_i \rightarrow B_j\) and \(\delta\) is a list of list of morphisms \(\delta_{ij}:C_j \rightarrow D_i\), where \(i = 1, \dots, m\) and \(j = 1, \dots, n\) for integers \(m,n \geq 1\). The method delegates to BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory applied on \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) where
\(\beta_{ij}\) equals IdentityMorphism(Source(\(\delta_{ij}\))) if \(\delta_{ij}\) is an endomorphism, and ZeroMorphism(Source(\(\delta_{ij}\)), Range(\(\delta_{ij}\))) otherwise,
\(\gamma_{ij}\) equals IdentityMorphism(Source(\(\alpha_{ij}\))) if \(\alpha_{ij}\) is an endomorphism, and ZeroMorphism(Source(\(\alpha_{ij}\)), Range(\(\alpha_{ij}\))) otherwise;
for all \(i = 1, \dots, m\) and \(j = 1, \dots, n\).
gap> LoadPackage( "ToolsForCategoricalTowers" ); true gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> QQ := HomalgFieldOfRationals(); Q gap> QQ_rows := CategoryOfRows( QQ ); Rows( Q ) gap> t := TensorUnit( QQ_rows ); <A row module over Q of rank 1> gap> id_t := IdentityMorphism( t ); <An identity morphism in Rows( Q )> gap> 1*(11)*7 + 2*(12)*8 + 3*(13)*9; 620 gap> 4*(11)*3 + 5*(12)*4 + 6*(13)*1; 450 gap> alpha_s := [ [ 1 , 2 , 3 ], [ 4 , 5 , 6 ] ] * id_t;; gap> beta_s := [ [ 7 , 8 , 9 ], [ 3 , 4 , 1 ] ] * id_t;; gap> gamma_s := [ 620 , 450 ] * id_t;; gap> MereExistenceOfSolutionOfLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s, gamma_s ); true gap> MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s, gamma_s ); false gap> x := SolveLinearSystemInAbCategory( QQ_rows, alpha_s, beta_s, gamma_s );; gap> 1*x[1]*7 + 2*x[2]*8 + 3*x[3]*9 = gamma_s[1]; true gap> 4*x[1]*3 + 5*x[2]*4 + 6*x[3]*1 = gamma_s[2]; true gap> MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s ); false gap> basis := BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory > ( QQ_rows, alpha_s, beta_s );; gap> Length( basis ); 1 gap> Display( basis[1][1] ); Source: A row module over Q of rank 1 Matrix: [ [ 111/13 ] ] Range: A row module over Q of rank 1 A morphism in Rows( Q ) gap> Display( basis[1][2] ); Source: A row module over Q of rank 1 Matrix: [ [ -141/26 ] ] Range: A row module over Q of rank 1 A morphism in Rows( Q ) gap> Display( basis[1][3] ); Source: A row module over Q of rank 1 Matrix: [ [ 1 ] ] Range: A row module over Q of rank 1 A morphism in Rows( Q ) gap> 2*(11)*5 + 3*(12)*7 + 9*(13)*2; 596 gap> Add( alpha_s, [ 2 , 3 , 9 ] * id_t ); gap> Add( beta_s, [ 5 , 7 , 2 ] * id_t ); gap> Add( gamma_s, 596 * id_t ); gap> MereExistenceOfSolutionOfLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s, gamma_s ); true gap> MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s, gamma_s ); true gap> x := SolveLinearSystemInAbCategory( QQ_rows, alpha_s, beta_s, gamma_s );; gap> 1*x[1]*7 + 2*x[2]*8 + 3*x[3]*9 = gamma_s[1]; true gap> 4*x[1]*3 + 5*x[2]*4 + 6*x[3]*1 = gamma_s[2]; true gap> 2*x[1]*5 + 3*x[2]*7 + 9*x[3]*2 = gamma_s[3]; true gap> MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory > ( QQ_rows, alpha_s, beta_s ); true gap> basis := BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory > ( QQ_rows, alpha_s, beta_s ); [ ]
‣ LimitPair( cat, objs, decorated_mors ) | ( operation ) |
Returns: a pair of morphisms
The input is
a category cat and
a finite diagram (or subquiver) in cat given by
a list objs of objects in cat, and
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat.
The output is a pair consisting of an object in cat and two parallel morphisms having this object as source, such that the binary equalizer of the two morphisms is the limit of the diagram.
‣ Limit( objs, decorated_mors ) | ( operation ) |
Returns: an object
The input is a finite diagram (or subquiver) D (in a category cat) consisting of
a list objs of objects in cat and
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat.
The output is the limit object \(\mathrm{Limit}\)(D) of the diagram D.
‣ ProjectionInFactorOfLimit( objects, decorated_morphisms, k ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, P_k )\)
The input is
a finite diagram (or subquiver) D (in a category cat) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in cat,
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat, and
a positive integer k.
The output is the projection of the limit object \(P := \mathrm{Limit}\)(D) of the diagram D into the k-th objects \(P_k =\)objs[k].
‣ ProjectionInFactorOfLimitWithGivenLimit( objects, decorated_morphisms, k, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, P_k )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category cat) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in cat,
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat,
a positive integer k, and
an object P\(= \mathrm{Limit}(D)\).
The output is the projection of the limit object P \(:=\mathrm{Limit}(D)\) into \(P_k =\)objs[k].
‣ UniversalMorphismIntoLimit( objects, decorated_morphisms, T, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, \mathrm{Limit}(D) )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category \(cat\)) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in \(cat\),
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in \(cat\),
a test object T, and
a list of morphisms \(\tau = ( \tau_i: T \rightarrow P_i )\) in \(cat\) defining a cone over \(D\).
The output is the morphism \(u( \tau ): T \rightarrow \mathrm{Limit}(D)\) given by the universal property of the limit.
‣ UniversalMorphismIntoLimitWithGivenLimit( objects, decorated_morphisms, T, tau, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, P )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category \(cat\)) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in \(cat\),
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in \(cat\),
a test object T,
a list of morphisms \(\tau = ( \tau_i: T \rightarrow P_i )\) in \(cat\) defining a cone over \(D\), and
an object P\(= \mathrm{Limit}(D)\).
The output is the morphism \(u( \tau ): T \rightarrow P = \mathrm{Limit}(D)\) given by the universal property of the limit.
‣ ColimitPair( cat, objs, decorated_mors ) | ( operation ) |
Returns: a pair of morphisms
The input is
a category cat and
a finite diagram (or subquiver) in cat given by
a list objs of objects in cat, and
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat.
The output is a pair consisting of an object in cat and two parallel morphisms having this object as range, such that the binary coequalizer of the two morphisms is the colimit of the diagram.
‣ Colimit( objs, decorated_mors ) | ( operation ) |
Returns: an object
The input is a finite diagram (or subquiver) D (in a category cat) consisting of
a list objs of objects in cat and
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat.
The output is the colimit object \(\mathrm{Colimit}\)(D) of the diagram D.
‣ InjectionOfCofactorOfColimit( objects, decorated_morphisms, k ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P_k, P )\)
The input is
a finite diagram (or subquiver) D (in a category cat) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in cat,
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat, and
a positive integer k.
The output is the injection of the k-th objects \(P_k =\)objs[k] into the colimit object \(P := \mathrm{Colimit}\)(D) of the diagram D.
‣ InjectionOfCofactorOfColimitWithGivenColimit( objects, decorated_morphisms, k, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P_k, P )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category cat) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in cat,
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in cat,
a positive integer k, and
an object P\(= \mathrm{Colimit}(D)\).
The output is the projection of \(P_k =\)objs[k] into the colimit object P \(:=\mathrm{Colimit}(D)\).
‣ UniversalMorphismFromColimit( objects, decorated_morphisms, T, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Colimit}(D), T )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category \(cat\)) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in \(cat\),
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in \(cat\),
a test object T, and
a list of morphisms \(\tau = ( \tau_i: P_i \rightarrow T )\) in \(cat\) defining a cocone over \(D\).
The output is the morphism \(u( \tau ): \mathrm{Colimit}(D) \rightarrow T\) given by the universal property of the colimit.
‣ UniversalMorphismFromColimitWithGivenColimit( objects, decorated_morphisms, T, tau, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, T )\)
The input is
a finite diagram (or subquiver) \(D\) (in a category \(cat\)) consisting of
a list objs = \((P_i)_{i = 1 \dots n}\) of objects in \(cat\),
a list decorated_mors of triples [ i, mor, j ], where mor:objs[i] \(\to\) objs[j] is a morphism in \(cat\),
a test object T,
a list of morphisms \(\tau = ( \tau_i: P_i \rightarrow T )\) in \(cat\) defining a cocone over \(D\), and
an object P\(= \mathrm{Colimit}(D)\).
The output is the morphism \(u( \tau ): P = \mathrm{Colimit}(D) \rightarrow T\) given by the universal property of the colimit.
‣ PositionsOfSublist( superlist, sublist[, from] ) | ( function ) |
Returns: a list of integers
returns all indices \(1 \leq i \leq\) Length(superlist) at which a sublist (of superlist) equal to sublist starts. If sublist does not occur the output is the empty-list. The version with given from starts searching after the position from.
‣ AllCoproducts( cat, objects ) | ( operation ) |
Returns: a list of objects
The input is a category cat and a list of objects therein, such that the list \(L\) of all possible finite coproducts is finite. The output is the list \(L\).
‣ AddBasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory. \(F: ( arg2, arg3, arg4, arg5 ) \mapsto \mathtt{BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategory}(arg2, arg3, arg4, arg5)\).
‣ AddBasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory. \(F: ( arg2, arg3 ) \mapsto \mathtt{BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory}(arg2, arg3)\).
‣ AddBiasedRelativeWeakFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation BiasedRelativeWeakFiberProduct. \(F: ( arg2, arg3, arg4 ) \mapsto \mathtt{BiasedRelativeWeakFiberProduct}(arg2, arg3, arg4)\).
‣ AddColimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Colimit. \(F: ( objects, decorated_morphisms ) \mapsto \mathtt{Colimit}(objects, decorated_morphisms)\).
‣ AddEmbeddingOfSumOfImagesOfAllMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EmbeddingOfSumOfImagesOfAllMorphisms. \(F: ( a, b ) \mapsto \mathtt{EmbeddingOfSumOfImagesOfAllMorphisms}(a, b)\).
‣ AddInjectionOfCofactorOfColimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfColimit. \(F: ( objects, decorated_morphisms, k ) \mapsto \mathtt{InjectionOfCofactorOfColimit}(objects, decorated_morphisms, k)\).
‣ AddInjectionOfCofactorOfColimitWithGivenColimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfColimitWithGivenColimit. \(F: ( objects, decorated_morphisms, k, P ) \mapsto \mathtt{InjectionOfCofactorOfColimitWithGivenColimit}(objects, decorated_morphisms, k, P)\).
‣ AddIsWeakInitial( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsWeakInitial. \(F: ( arg2 ) \mapsto \mathtt{IsWeakInitial}(arg2)\).
‣ AddIsWeakTerminal( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsWeakTerminal. \(F: ( arg2 ) \mapsto \mathtt{IsWeakTerminal}(arg2)\).
‣ AddLimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Limit. \(F: ( objects, decorated_morphisms ) \mapsto \mathtt{Limit}(objects, decorated_morphisms)\).
‣ AddMereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory. \(F: ( arg2, arg3 ) \mapsto \mathtt{MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCategory}(arg2, arg3)\).
‣ AddMereExistenceOfUniqueSolutionOfLinearSystemInAbCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory. \(F: ( arg2, arg3, arg4 ) \mapsto \mathtt{MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory}(arg2, arg3, arg4)\).
‣ AddMorphismBetweenCoproducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismBetweenCoproducts. \(F: ( source_diagram, pair, range_diagram ) \mapsto \mathtt{MorphismBetweenCoproducts}(source_diagram, pair, range_diagram)\).
‣ AddMorphismBetweenCoproductsWithGivenCoproducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismBetweenCoproductsWithGivenCoproducts. \(F: ( S, source_diagram, pair, range_diagram, T ) \mapsto \mathtt{MorphismBetweenCoproductsWithGivenCoproducts}(S, source_diagram, pair, range_diagram, T)\).
‣ AddMorphismBetweenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismBetweenDirectProducts. \(F: ( source_diagram, pair, range_diagram ) \mapsto \mathtt{MorphismBetweenDirectProducts}(source_diagram, pair, range_diagram)\).
‣ AddMorphismBetweenDirectProductsWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismBetweenDirectProductsWithGivenDirectProducts. \(F: ( S, source_diagram, pair, range_diagram, T ) \mapsto \mathtt{MorphismBetweenDirectProductsWithGivenDirectProducts}(S, source_diagram, pair, range_diagram, T)\).
‣ AddMorphismOntoSumOfImagesOfAllMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismOntoSumOfImagesOfAllMorphisms. \(F: ( a, b ) \mapsto \mathtt{MorphismOntoSumOfImagesOfAllMorphisms}(a, b)\).
‣ AddProjectionInFactorOfLimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfLimit. \(F: ( objects, decorated_morphisms, k ) \mapsto \mathtt{ProjectionInFactorOfLimit}(objects, decorated_morphisms, k)\).
‣ AddProjectionInFactorOfLimitWithGivenLimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfLimitWithGivenLimit. \(F: ( objects, decorated_morphisms, k, P ) \mapsto \mathtt{ProjectionInFactorOfLimitWithGivenLimit}(objects, decorated_morphisms, k, P)\).
‣ AddProjectionOfBiasedRelativeWeakFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionOfBiasedRelativeWeakFiberProduct. \(F: ( a, b, c ) \mapsto \mathtt{ProjectionOfBiasedRelativeWeakFiberProduct}(a, b, c)\).
‣ AddProjectionOfBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionOfBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct. \(F: ( a, b, c, P ) \mapsto \mathtt{ProjectionOfBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct}(a, b, c, P)\).
‣ AddRelativeLift( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation RelativeLift. \(F: ( beta, alpha, nu ) \mapsto \mathtt{RelativeLift}(beta, alpha, nu)\).
‣ AddSetOfMorphismsOfFiniteCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation SetOfMorphismsOfFiniteCategory. \(F: ( ) \mapsto \mathtt{SetOfMorphismsOfFiniteCategory}()\).
‣ AddSetOfObjectsOfCategory( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation SetOfObjectsOfCategory. \(F: ( ) \mapsto \mathtt{SetOfObjectsOfCategory}()\).
‣ AddSumOfImagesOfAllMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation SumOfImagesOfAllMorphisms. \(F: ( arg2, arg3 ) \mapsto \mathtt{SumOfImagesOfAllMorphisms}(arg2, arg3)\).
‣ AddUniversalMorphismFromColimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromColimit. \(F: ( objects, decorated_morphisms, T, tau ) \mapsto \mathtt{UniversalMorphismFromColimit}(objects, decorated_morphisms, T, tau)\).
‣ AddUniversalMorphismFromColimitWithGivenColimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromColimitWithGivenColimit. \(F: ( objects, decorated_morphisms, T, tau, P ) \mapsto \mathtt{UniversalMorphismFromColimitWithGivenColimit}(objects, decorated_morphisms, T, tau, P)\).
‣ AddUniversalMorphismIntoBiasedRelativeWeakFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoBiasedRelativeWeakFiberProduct. \(F: ( a, b, c, t ) \mapsto \mathtt{UniversalMorphismIntoBiasedRelativeWeakFiberProduct}(a, b, c, t)\).
‣ AddUniversalMorphismIntoBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct. \(F: ( a, b, c, t, P ) \mapsto \mathtt{UniversalMorphismIntoBiasedRelativeWeakFiberProductWithGivenBiasedRelativeWeakFiberProduct}(a, b, c, t, P)\).
‣ AddUniversalMorphismIntoLimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoLimit. \(F: ( objects, decorated_morphisms, T, tau ) \mapsto \mathtt{UniversalMorphismIntoLimit}(objects, decorated_morphisms, T, tau)\).
‣ AddUniversalMorphismIntoLimitWithGivenLimit( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoLimitWithGivenLimit. \(F: ( objects, decorated_morphisms, T, tau, P ) \mapsto \mathtt{UniversalMorphismIntoLimitWithGivenLimit}(objects, decorated_morphisms, T, tau, P)\).
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