GAP (FinGSetsForCAP) - Chapter 3: The skeletal category of transitive left $G$-sets

The group $G$ underlying the skeletal category C of transitive left $G$-set.

3.1-2 UnderlyingGroupAsCategory

The group $G$ underlying the skeletal category C of transitive left $G$-set, viewed as a category on one object.

3.1-3 UnderlyingTableOfMarks

The table of marks of the group $G$ underlying the skeletal category C of transitive left $G$-set.

3.1-4 NumberOfObjects

The number of objects of the skeletal category C of transitive left $G$-set.

3.1-5 CardinalitiesOfObjects

The list of cardinalities of objects of the skeletal category C of transitive left $G$-set.

3.1-6 RepresentativesOfSubgroupsUpToConjugation

The number of objects of the skeletal category C of transitive left $G$-set.

3.1-7 ObjectNumber

The positive integer $i$ such that the transitive left $G$-set Omega $\cong U_i \backslash G$, i.e., ObjectNumber( TransitiveGSet( G, i ) ) = i.

3.1-8 UnderlyingGroupElement

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, $g U_s \subseteq U_t g$.

3.1-9 Cardinality

3.1-10 Size

3.1-11 CoequalizerMorphisms

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 $\cong G / U_i$.

3.1-12 EmbeddingOfUnderlyingGroupAsCategory

The full embedding functor from the group $G$ underlying the finite coequalizer completion TG into TG.

3.1-13 ExtendFunctorToSkeletalCategoryOfTransitiveLeftGSets

The full embedding functor from the group $G$ underlying the finite coequalizer completion TG into TG.

3.2 Constructors

3.2-1 SkeletalCategoryOfTransitiveLeftGSets

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

3.3 GAP Categories

3.4 GAP categories

3.4-1 IsSkeletalCategoryOfTransitiveLeftGSets

3.4-2 IsObjectInSkeletalCategoryOfTransitiveLeftGSets

The GAP category of objects in the skeletal category of transitive left $G$-sets.

3.4-3 IsMorphismInSkeletalCategoryOfTransitiveLeftGSets

The GAP category of morphisms in the skeletal category of transitive left $G$-sets.

3 The skeletal category of transitive left $G$-sets

3.1 Attributes

3.1-1 UnderlyingGroup