‣ DataTables( C ) | ( attribute ) |
Returns: a pair of lists
The data tables used to create the category C. It might differ from the normalized
output of DataTablesOfCategory( C ).
‣ Size( C ) | ( attribute ) |
Returns: a nonnegative integer
The number of morphisms in the category C created from data tables.
‣ OppositeCategoryFromDataTables( C ) | ( attribute ) |
Returns: a CAP category
The opposite category of the category C defined by nerve data.
‣ CategoryFromDataTables( input_record ) | ( attribute ) |
Returns: a CAP category
Construct an enriched finite category out of the input_record consisting of values to the keys:
name
nerve_data
indices_of_generating_morphisms
relations
labels
properties
gap> LoadPackage( "Algebroids", false ); true gap> Delta1 := SimplicialCategoryTruncatedInDegree( 1 ); FreeCategory( RightQuiver( "Delta(C0,C1)[id:C1->C0,s:C0->C1,t:C0->C1]" ) ) / [ s*id = C0, t*id = C0 ] gap> Size( Delta1 ); 7 gap> mors := SetOfMorphisms( Delta1 ); [ (C0)-[(C0)]->(C0), (C1)-[(id)]->(C0), (C0)-[(s)]->(C1), (C0)-[(t)]->(C1), (C1)-[(C1)]->(C1), (C1)-[(id*s)]->(C1), (C1)-[(id*t)]->(C1) ] gap> List( mors, DecompositionOfMorphismInCategory ); [ [ ], [ (C1)-[(id)]->(C0) ], [ (C0)-[(s)]->(C1) ], [ (C0)-[(t)]->(C1) ], [ ], [ (C1)-[(id)]->(C0), (C0)-[(s)]->(C1) ], [ (C1)-[(id)]->(C0), (C0)-[(t)]->(C1) ] ] gap> C := CategoryFromDataTables( Delta1 ); FreeCategory( RightQuiver( "Delta(C0,C1)[id:C1->C0,s:C0->C1,t:C0->C1]" ) ) / [ s*id = C0, t*id = C0 ] gap> Size( C ); 7 gap> morsC := SetOfMorphisms( C ); [ (C0)-[(C0)]->(C0), (C1)-[(id)]->(C0), (C0)-[(s)]->(C1), (C0)-[(t)]->(C1), (C1)-[(C1)]->(C1), (C1)-[(id*s)]->(C1), (C1)-[(id*t)]->(C1) ] gap> List( morsC, DecompositionOfMorphismInCategory ); [ [ ], [ (C1)-[(id)]->(C0) ], [ (C0)-[(s)]->(C1) ], [ (C0)-[(t)]->(C1) ], [ ], [ (C1)-[(id)]->(C0), (C0)-[(s)]->(C1) ], [ (C1)-[(id)]->(C0), (C0)-[(t)]->(C1) ] ] gap> NerveTruncatedInDegree2Data( C ) = NerveTruncatedInDegree2Data( Delta1 ); true gap> IndicesOfGeneratingMorphisms( C ); [ 1, 2, 3 ] gap> SetOfGeneratingMorphisms( C ); [ (C1)-[(id)]->(C0), (C0)-[(s)]->(C1), (C0)-[(t)]->(C1) ] gap> Display( C ); A CAP category with name FreeCategory( RightQuiver( "Delta(C0,C1)[id:C1->C0,s:C0->C1,t:C0->C1]" ) ) / [ s*id = C0, t*id = C0 ]: 19 primitive operations were used to derive 57 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsFiniteCategory * IsEquippedWithHomomorphismStructure gap> C0 := CreateObject( C, 0 ); <(C0)> gap> IsWellDefined( C0 ); true gap> C1 := 1 / C; <(C1)> gap> IsWellDefined( C1 ); true gap> IsWellDefined( 2 / C ); false gap> idC0 := CreateMorphism( C0, 0, C0 ); (C0)-[(C0)]->(C0) gap> CreateMorphism( C, 0 ) = idC0; true gap> IsOne( idC0 ); true gap> id := CreateMorphism( C, 1 ); (C1)-[(id)]->(C0) gap> s := CreateMorphism( C, 2 ); (C0)-[(s)]->(C1) gap> t := CreateMorphism( C, 3 ); (C0)-[(t)]->(C1) gap> IsSplitMonomorphism( s ); true gap> IsSplitMonomorphism( t ); true gap> IsEpimorphism( s ); false gap> IsEpimorphism( t ); false gap> IsSplitEpimorphism( id ); true gap> IsMonomorphism( id ); false gap> idC1 := CreateMorphism( C, 4 ); (C1)-[(C1)]->(C1) gap> IsOne( idC1 ); true gap> sigma := CreateMorphism( C, 5 ); (C1)-[(id*s)]->(C1) gap> tau := CreateMorphism( C, 6 ); (C1)-[(id*t)]->(C1) gap> IsEndomorphism( sigma ); true gap> IsMonomorphism( sigma ); false gap> IsEpimorphism( sigma ); false gap> IsEndomorphism( tau ); true gap> IsMonomorphism( tau ); false gap> IsEpimorphism( tau ); false gap> IsOne( tau ); false gap> IsWellDefined( CreateMorphism( C1, 7, C1 ) ); false gap> PreCompose( s, id ) = idC0; true gap> PreCompose( t, id ) = idC0; true gap> PreCompose( id, s ) = sigma; true gap> PreCompose( id, t ) = tau; true gap> HomStructure( C0, C0 ); |1| gap> HomStructure( C1, C1 ); |3| gap> HomStructure( C0, C1 ); |2| gap> HomStructure( C1, C0 ); |1| gap> Display( HomStructure( s ) ); { 0 } ⱶ[ 0 ]→ { 0, 1 } gap> Display( HomStructure( t ) ); { 0 } ⱶ[ 1 ]→ { 0, 1 } gap> HomStructure( Source( s ), Target( s ), HomStructure( s ) ) = s; true gap> HomStructure( Source( t ), Target( t ), HomStructure( t ) ) = t; true gap> Display( HomStructure( s, t ) ); { 0 } ⱶ[ 1 ]→ { 0, 1 } gap> Display( HomStructure( t, s ) ); { 0 } ⱶ[ 0 ]→ { 0, 1 } gap> Display( HomStructure( sigma, tau ) ); { 0, 1, 2 } ⱶ[ 2, 2, 2 ]→ { 0, 1, 2 } gap> Display( HomStructure( > PreCompose( Delta1.id, Delta1.s ), > PreCompose( Delta1.id, Delta1.t ) ) ); { 0, 1, 2 } ⱶ[ 2, 2, 2 ]→ { 0, 1, 2 } gap> Display( HomStructure( tau, sigma ) ); { 0, 1, 2 } ⱶ[ 1, 1, 1 ]→ { 0, 1, 2 } gap> Display( HomStructure( > PreCompose( Delta1.id, Delta1.t ), > PreCompose( Delta1.id, Delta1.s ) ) ); { 0, 1, 2 } ⱶ[ 1, 1, 1 ]→ { 0, 1, 2 } gap> Display( HomStructure( tau, idC1 ) ); { 0, 1, 2 } ⱶ[ 2, 1, 2 ]→ { 0, 1, 2 } gap> Display( HomStructure( idC1, idC1 ) ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0, 1, 2 } gap> mors := SetOfMorphisms( C ); [ (C0)-[(C0)]->(C0), (C1)-[(id)]->(C0), (C0)-[(s)]->(C1), (C0)-[(t)]->(C1), (C1)-[(C1)]->(C1), (C1)-[(id*s)]->(C1), (C1)-[(id*t)]->(C1) ] gap> List( mors, OppositeMorphismInOppositeCategoryFromDataTables ); [ (C0)-[(C0)]->(C0), (C0)-[(id)]->(C1), (C1)-[(s)]->(C0), (C1)-[(t)]->(C0), (C1)-[(C1)]->(C1), (C1)-[(s*id)]->(C1), (C1)-[(t*id)]->(C1) ] gap> mors; [ (C0)-[(C0)]->(C0), (C1)-[(id)]->(C0), (C0)-[(s)]->(C1), (C0)-[(t)]->(C1), (C1)-[(C1)]->(C1), (C1)-[(id*s)]->(C1), (C1)-[(id*t)]->(C1) ] gap> List( mors, DecompositionOfMorphismInCategory ); [ [ ], [ (C1)-[(id)]->(C0) ], [ (C0)-[(s)]->(C1) ], [ (C0)-[(t)]->(C1) ], [ ], [ (C1)-[(id)]->(C0), (C0)-[(s)]->(C1) ], [ (C1)-[(id)]->(C0), (C0)-[(t)]->(C1) ] ] gap> C_op := OppositeCategoryFromDataTables( C ); Opposite( FreeCategory( RightQuiver( "Delta(C0,C1)[id:C1->C0,s:C0->C1,t:C0->C1]" ) ) / [ s*id = C0, t*id = C0 ] ) gap> IsIdenticalObj( OppositeCategoryFromDataTables( C_op ), C ); true gap> IndicesOfGeneratingMorphisms( C_op ); [ 3, 1, 2 ] gap> SetOfGeneratingMorphisms( C_op ); [ (C0)-[(id)]->(C1), (C1)-[(s)]->(C0), (C1)-[(t)]->(C0) ] gap> mors_op := SetOfMorphisms( C_op ); [ (C0)-[(C0)]->(C0), (C1)-[(s)]->(C0), (C1)-[(t)]->(C0), (C0)-[(id)]->(C1), (C1)-[(C1)]->(C1), (C1)-[(s*id)]->(C1), (C1)-[(t*id)]->(C1) ] gap> List( mors_op, DecompositionOfMorphismInCategory ); [ [ ], [ (C1)-[(s)]->(C0) ], [ (C1)-[(t)]->(C0) ], [ (C0)-[(id)]->(C1) ], [ ], [ (C1)-[(s)]->(C0), (C0)-[(id)]->(C1) ], [ (C1)-[(t)]->(C0), (C0)-[(id)]->(C1) ] ]
‣ CategoryFromDataTables( C ) | ( attribute ) |
‣ CategoryFromDataTables( C ) | ( attribute ) |
‣ CreateObject( C, o ) | ( operation ) |
Returns: a CAP category
Construct the o-th object in the category C created from data tables.
‣ CreateMorphism( C, m ) | ( operation ) |
‣ CreateMorphism( source, m, range ) | ( operation ) |
‣ CreateMorphism( C, m ) | ( operation ) |
‣ CreateMorphism( source, m, range ) | ( operation ) |
Returns: a CAP category
Construct the m-th morphism source\(\to\)range in the category C created from data tables.
Construct the m-th morphism source\(\to\)range in the category C created from nerve data.
‣ IsCategoryFromDataTables( arg ) | ( filter ) |
Returns: true or false
The GAP category of categories from data tables.
‣ IsCellInCategoryFromDataTables( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a category from data tables.
‣ IsObjectInCategoryFromDataTables( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in a category from data tables.
‣ IsMorphismInCategoryFromDataTables( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a category from data tables.
generated by GAPDoc2HTML