‣ UnderlyingGroup( C ) | ( attribute ) |
Returns: a group
The group G underlying the skeletal category C of transitive left G-set.
‣ UnderlyingGroupAsCategory( C ) | ( attribute ) |
Returns: a group
The group G underlying the skeletal category C of transitive left G-set, viewed as a category on one object.
‣ UnderlyingTableOfMarks( C ) | ( attribute ) |
Returns: a table of marks
The table of marks of the group G underlying the skeletal category C of transitive left G-set.
‣ NumberOfObjects( C ) | ( attribute ) |
Returns: a positive integer
The number of objects of the skeletal category C of transitive left G-set.
‣ CardinalitiesOfObjects( C ) | ( attribute ) |
Returns: a positive integer
The list of cardinalities of objects of the skeletal category C of transitive left G-set.
‣ RepresentativesOfSubgroupsUpToConjugation( C ) | ( attribute ) |
Returns: a positive integer
The number of objects of the skeletal category C of transitive left G-set.
‣ ObjectNumber( Omega ) | ( attribute ) |
Returns: a positive integer
The positive integer i such that the transitive left G-set Omega \cong U_i \backslash G, i.e., ObjectNumber( TransitiveGSet( G, i ) ) = i.
‣ UnderlyingGroupElement( phi ) | ( attribute ) |
Returns: a group element
The group elements g \in G defining the morphism phi: \cong U_s \backslash G \to U_t \backslash G satisfying g U_s g^{-1} \leq U_t, or, equivalently, g U_s \subseteq U_t g.
‣ Cardinality( Omega ) | ( attribute ) |
Returns: a positive integer
The cardinality of the transitive left G-set Omega.
‣ Size( Omega ) | ( attribute ) |
Returns: a positive integer
The cardinality of the transitive left G-set Omega.
‣ CoequalizerMorphisms( Omega ) | ( attribute ) |
Returns: a list of morphisms
The input is a transitive left G-set Omega \cong G / U_i. The output is the list of endomorphisms of the principal G-set G / U_1 defined by the generators of U_i. The coequalizers of these morphisms is the input object Omega \cong G / U_i.
‣ EmbeddingOfUnderlyingGroupAsCategory( UC ) | ( attribute ) |
Returns: a CAP functor
The full embedding functor from the group G underlying the finite coequalizer completion TG into TG.
‣ ExtendFunctorToSkeletalCategoryOfTransitiveLeftGSets( UC ) | ( attribute ) |
Returns: a CAP functor
The full embedding functor from the group G underlying the finite coequalizer completion TG into TG.
‣ SkeletalCategoryOfTransitiveLeftGSets( G ) | ( attribute ) |
Returns: a category
The argument is a group G. The output is the skeletal category of transitive G-Sets.
gap> LoadPackage( "FinGSetsForCAP", false ); true gap> LoadPackage( "Locales", false ); true gap> S3 := SymmetricGroup( 3 );; StructureDescription( S3 );; S3; S3 gap> TS3 := SkeletalCategoryOfTransitiveLeftGSets( S3 ); SkeletalCategoryOfTransitiveLeftGSets( S3 ) with 4 objects gap> Display( TS3 ); A CAP category with name SkeletalCategoryOfTransitiveLeftGSets( S3 ) \ with 4 objects: 24 primitive operations were used to derive 71 operations for this category \ which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsCategoryWithTerminalObject * IsFiniteCategory * IsCategoryWithCoequalizers * IsEquippedWithHomomorphismStructure and not yet algorithmically * IsFinitelyPresentedCategory and furthermore mathematically * IsSkeletalCategory gap> Omega1 := TS3.1; S3 / U_1 gap> Omega2 := TS3.2; S3 / U_2 gap> Omega3 := TS3.3; S3 / U_3 gap> Omega4 := TS3.4; S3 / U_4 gap> objs := SetOfObjects( TS3 ); [ S3 / U_1, S3 / U_2, S3 / U_3, S3 / U_4 ] gap> List( objs, Size ); [ 6, 3, 2, 1 ] gap> t := TerminalObject( TS3 ); S3 / U_4 gap> ObjectDatum( t ); 4 gap> u := UniversalMorphismIntoTerminalObject( Omega2 ); (): S3 / U_2 -> S3 / U_4 gap> MorphismDatum( u ); () gap> mors := SetOfMorphisms( TS3 ); [ (): S3 / U_1 -> S3 / U_1, (2,3): S3 / U_1 -> S3 / U_1,\ (1,2): S3 / U_1 -> S3 / U_1, (1,3,2): S3 / U_1 -> S3 / U_1,\ (1,2,3): S3 / U_1 -> S3 / U_1, (1,3): S3 / U_1 -> S3 / U_1,\ (): S3 / U_1 -> S3 / U_2, (1,3): S3 / U_1 -> S3 / U_2,\ (1,3,2): S3 / U_1 -> S3 / U_2, (): S3 / U_2 -> S3 / U_2,\ (): S3 / U_1 -> S3 / U_3, (2,3): S3 / U_1 -> S3 / U_3,\ (): S3 / U_3 -> S3 / U_3, (2,3): S3 / U_3 -> S3 / U_3,\ (): S3 / U_1 -> S3 / U_4, (): S3 / U_2 -> S3 / U_4,\ (): S3 / U_3 -> S3 / U_4, (): S3 / U_4 -> S3 / U_4 ] gap> ForAll( mors, IsWellDefined ); true gap> ForAll( mors, IsEpimorphism ); true gap> List( mors, IsEndomorphism ); [ true, true, true, true, true, true, false, false, false,\ true, false, false, true, true, false, false, false, true ] gap> ForAll( mors, mor -> IsMonomorphism( mor ) = IsEndomorphism( mor ) ); true gap> ForAll( mors, mor -> IsSplitMonomorphism( mor ) = IsEndomorphism( mor ) ); true gap> ForAll( mors, mor -> IsSplitEpimorphism( mor ) = IsEndomorphism( mor ) ); true gap> ForAll( mors, mor -> IsIsomorphism( mor ) = IsEndomorphism( mor ) ); true gap> phi := MorphismConstructor( Omega1, (1,2), Omega1 ); (1,2): S3 / U_1 -> S3 / U_1 gap> phi = mors[3]; true gap> psi := MorphismConstructor( Omega1, (2,3), Omega1 ); (2,3): S3 / U_1 -> S3 / U_1 gap> psi = mors[2]; true gap> dp := MorphismDatum( PreCompose( phi, psi ) ); (1,3,2) gap> pd := MorphismDatum( phi ) * MorphismDatum( psi ); (1,3,2) gap> dp = pd; true gap> digraph := DigraphOfPoset( PosetOfCategory( TS3 ) ); <immutable digraph with 4 vertices, 4 edges> gap> Print( DotVertexLabelledDigraph( digraph ) ); //dot digraph subobject_lattice{ rankdir="BT" minlen=0 node [shape=circle width=0 height=0 fontsize=12 margin=0.01 \ fontname="DejaVu Serif,serif"] edge [arrowsize=0.5] 1 [label="1"] 2 [label="2"] 3 [label="3"] 4 [label="4"] 1 -> 2 1 -> 3 2 -> 4 3 -> 4 } gap> ForAll( objs, o -> o = Coequalizer( Omega1, CoequalizerMorphisms( o ) ) ); true
‣ IsSkeletalCategoryOfTransitiveLeftGSets( object ) | ( filter ) |
Returns: true or false
The GAP category of the skeletal category of transitive left G-sets.
‣ IsObjectInSkeletalCategoryOfTransitiveLeftGSets( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the skeletal category of transitive left G-sets.
‣ IsMorphismInSkeletalCategoryOfTransitiveLeftGSets( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the skeletal category of transitive left G-sets.
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