‣ CAP_INTERNAL_METHOD_NAME_LIST_FOR_PREORDERED_SET_OF_CATEGORY | ( global variable ) |
‣ AmbientCategory( P ) | ( attribute ) |
Returns: a list
The ambient category of the proset P.
‣ UnderlyingCell( cell ) | ( attribute ) |
Returns: a CAP cell
The cell in the ambient category underlying cell.
‣ CreateProsetOrPosetOfCategory( C ) | ( operation ) |
‣ ProsetOfCategory( C ) | ( attribute ) |
‣ StableProsetOfCategory( C ) | ( attribute ) |
‣ PosetOfCategory( C ) | ( attribute ) |
‣ StablePosetOfCategory( C ) | ( attribute ) |
‣ AsCellOfProset( c ) | ( attribute ) |
‣ AsCellOfStableProset( c ) | ( attribute ) |
‣ AsCellOfPoset( c ) | ( attribute ) |
‣ AsCellOfStablePoset( c ) | ( attribute ) |
‣ /( object, P ) | ( operation ) |
gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" ); true gap> LoadPackage( "LazyCategories", ">= 2020.05.01" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x,y"]; Q[x,y] gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> I := KroneckerMat( K, J ); <An unevaluated 4 x 1 matrix over an external ring> gap> P := CategoryOfRows( R ); Rows( Q[x,y] ) gap> P := LazyCategory( P : show_evaluation := false ); LazyCategory( Rows( Q[x,y] ) ) gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> R := Range( I ); SetLabel( R, "R" ); <An evaluated object in LazyCategory( Rows( Q[x,y] ) )> gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K := K / P; <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> S := SliceCategoryOverTensorUnit( P ); SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) gap> L := PosetOfCategory( S ); Poset( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) gap> I := I / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> J := J / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K := K / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J An object in the poset given by: An object in the slice category given by: <A morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K = IqJ; true
gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" ); true gap> LoadPackage( "LazyCategories", ">= 2020.05.01" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x,y"]; Q[x,y] gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> I := KroneckerMat( K, J ); <An unevaluated 4 x 1 matrix over an external ring> gap> P := CategoryOfRows( R ); Rows( Q[x,y] ) gap> P := LazyCategory( P : primitive_operations := true, show_evaluation := false ); LazyCategory( Rows( Q[x,y] ) ) gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> R := Range( I ); SetLabel( R, "R" ); <An evaluated object in LazyCategory( Rows( Q[x,y] ) )> gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K := K / P; <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> S := SliceCategoryOverTensorUnit( P ); SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) gap> L := PosetOfCategory( S ); Poset( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) gap> I := I / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> J := J / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K := K / S / L; An object in the poset given by: An object in the slice category given by: <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J An object in the poset given by: An object in the slice category given by: <A morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K = IqJ; true
gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.04.16" ); true gap> LoadPackage( "LazyCategories", ">= 2020.06.01" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x,y"]; Q[x,y] gap> K := HomalgMatrix( "[ x, -x ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ); <A 2 x 1 matrix over an external ring> gap> I := KroneckerMat( K, J ); <An unevaluated 4 x 1 matrix over an external ring> gap> P := CategoryOfRows( R ); Rows( Q[x,y] ) gap> P := LazyCategory( P : show_evaluation := false ); LazyCategory( Rows( Q[x,y] ) ) gap> I := I / P; SetLabel( I, "I" ); SetLabel( Source( I ), "R^i" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> R := Range( I ); SetLabel( R, "R" ); <An evaluated object in LazyCategory( Rows( Q[x,y] ) )> gap> J := J / P; SetLabel( J, "J" ); SetLabel( Source( J ), "R^j" ); <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> K := K / P; <An evaluated morphism in LazyCategory( Rows( Q[x,y] ) )> gap> S := LazyCategory( SliceCategoryOverTensorUnit( P ) ); LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) gap> L := LazyCategory( PosetOfCategory( S ) ); LazyCategory( Poset( LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) ) gap> I := I / S; SetLabel( I, "I" ); <An evaluated object in LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )> gap> I := I / L; SetLabel( I, "I" ); <An evaluated object in LazyCategory( Poset( LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )> gap> J := J / S; SetLabel( J, "J" ); <An evaluated object in LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )> gap> J := J / L; SetLabel( J, "J" ); <An evaluated object in LazyCategory( Poset( LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )> gap> K := K / S; SetLabel( K, "K" ); <An evaluated object in LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) )> gap> K := K / L; SetLabel( K, "K" ); <An evaluated object in LazyCategory( Poset( LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )> gap> IqJ := InternalHom( J, I ); ## this is the ideal quotient I : J <An object in LazyCategory( Poset( LazyCategory( SliceCategoryOverTensorUnit( LazyCategory( Rows( Q[x,y] ) ) ) ) ) )> gap> IsHomSetInhabited( K, IqJ ); true gap> IsHomSetInhabited( IqJ, K ); true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.10-02" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x,y"]; Q[x,y] gap> F := CategoryOfRows( R ); Rows( Q[x,y] ) gap> S := SliceCategoryOverTensorUnit( F ); SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) gap> P := PosetOfCategory( S ); Poset( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) gap> I := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P; An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ) / F / S / P; An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJ := TensorProduct( I, J ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IiJ := DirectProduct( I, J ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IiJ = I; true gap> IsHomSetInhabited( IJ, IiJ ); true gap> IsHomSetInhabited( IiJ, IJ ); false gap> IpJ := Coproduct( I, J ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IpJ = J; true gap> IJqJ := InternalHom( J, IJ ); ## this is the ideal quotient IJ : J An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJqJ = I; true gap> iota := InternalHom( UniversalMorphismIntoTerminalObject( J ), IJ ); <An epi-, monomorphism in Poset( SliceCategoryOverTensorUnit( \ Rows( Q[x,y] ) ) )> gap> IsWellDefined( iota ); true gap> IsIsomorphism( iota ); false gap> IJJ := TensorProduct( IJ, J ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJqJ := InternalHom( J, IJJ ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJqJ = I; false gap> IJJsJ := StableInternalHom( J, IJJ ); An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJsJ = I; true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.10-02" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x"] / "x^2"; Q[x]/( x^2 ) gap> F := CategoryOfRows( R ); Rows( Q[x]/( x^2 ) ) gap> S := SliceCategoryOverTensorUnit( F ); SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) ) gap> P := PosetOfCategory( S ); Poset( SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) ) ) gap> L := StablePosetOfCategory( P ); StablePoset( Poset( SliceCategoryOverTensorUnit( Rows( Q[x]/( x^2 ) ) ) ) ) gap> N := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P; An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x]/( x^2 ) )> gap> I := InitialObject( P ); An object in the poset given by: An object in the slice category given by: <A zero, split monomorphism in Rows( Q[x]/( x^2 ) )> gap> I = N; false gap> IsInitial( I ); true gap> IsInitial( N ); false gap> rN := N / L; An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x]/( x^2 ) )> gap> rI := I / L; An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A zero, split monomorphism in Rows( Q[x]/( x^2 ) )> gap> rI = rN; true gap> IsInitial( rI ); true gap> IsInitial( rN ); true
gap> LoadPackage( "SubcategoriesForCAP", ">= 2020.10-02" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q["x,y"]; Q[x,y] gap> F := CategoryOfRows( R ); Rows( Q[x,y] ) gap> S := SliceCategoryOverTensorUnit( F ); SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) gap> P := PosetOfCategory( S ); Poset( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) gap> L := StablePosetOfCategory( P ); StablePoset( Poset( SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) ) gap> I := HomalgMatrix( "[ x ]", 1, 1, R ) / F / S / P / L; An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> J := HomalgMatrix( "[ x, y ]", 2, 1, R ) / F / S / P / L; An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJ := TensorProduct( I, J ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IiJ := DirectProduct( I, J ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IsHomSetInhabited( IJ, IiJ ); true gap> IsHomSetInhabited( IiJ, IJ ); true gap> IiJ = I; true gap> IpJ := Coproduct( I, J ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IpJ = J; true gap> IJqJ := InternalHom( J, IJ ); ## this is the ideal quotient IJ : J An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJqJ = I; true gap> iota := InternalHom( UniversalMorphismIntoTerminalObject( J ), IJ ); <An epi-, monomorphism in StablePoset( Poset( \ SliceCategoryOverTensorUnit( Rows( Q[x,y] ) ) ) )> gap> IsWellDefined( iota ); true gap> IsOne( iota ); true gap> IJJ := TensorProduct( IJ, J ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJqJ := InternalHom( J, IJJ ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJqJ = I; true gap> IJJsJ := StableInternalHom( J, IJJ ); An object in the stable poset given by: An object in the poset given by: An object in the slice category given by: <A morphism in Rows( Q[x,y] )> gap> IJJsJ = I; true
‣ IsProsetOrPosetOfCapCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of a proset or poset of a CAP category.
‣ IsCapCategoryCellInProsetOrPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a proset or poset of a CAP category.
‣ IsCapCategoryCellInStableProsetOrPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a stable proset or poset of a CAP category.
‣ IsCapCategoryObjectInProsetOrPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in a proset or poset of a CAP category.
‣ IsProsetOfCapCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of a proset of a CAP category.
‣ IsCapCategoryCellInProsetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a proset of a CAP category.
‣ IsCapCategoryObjectInProsetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in a proset of a CAP category.
‣ IsCapCategoryMorphismInProsetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a proset of a CAP category.
‣ IsPosetOfCapCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of a poset of a CAP category.
‣ IsCapCategoryCellInPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a poset of a CAP category.
‣ IsCapCategoryObjectInPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in a poset of a CAP category.
‣ IsCapCategoryMorphismInPosetOfACategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a poset of a CAP category.
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