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5 Homomorphisms of Groupoids
 5.1 Homomorphisms to a connected groupoid
  5.1-1 GroupoidHomomorphism

5 Homomorphisms of Groupoids

A homomorphism m from a groupoid G to a groupoid H consists of a map from the objects of G to those of H together with a map from the elements of G to those of H which is compatible with tail and head and which preserves multiplication:

m(g1 : o1 \to o2)*m(g2 : o2 \to o3) ~=~ m(g1*g2 : o1 \to o3).

Note that when a homomorphism is not injective on objects, the image of the source need not be a subgroupoid of the range. The simplest example of this is given by homomorphism the two-object groupoid with trivial group to the free group < a > on one generator, when the image is [1,a,a^-1].

5.1 Homomorphisms to a connected groupoid

5.1-1 GroupoidHomomorphism
> GroupoidHomomorphism( args )( function )
> GroupoidHomomorphismFromSinglePiece( src, rng, hom, imobs )( operation )
> Source( hom )( attribute )
> Range( hom )( attribute )

As usual, there are a variety of homomorphism constructors. The basic construction is a homomorphism G -> H with H connected, which is implemented as IsHomomorphismToSinglePieceGroupoidRep with attributes Source, Range and PieceImages. If G is also connected, we may apply HomomorphismOfSinglePieceGroupoids, requiring:


gap> d12 := Group( (15,16,17,18,19,20, (15,20)(16,19)(17,18) );;
gap> Gd12 := SinglePieceGroupoid( [-37,-36,-35,-34], d12 );;
gap> SetName( d12, "d12" );  SetName( Gd12, "Gd12" );
gap> s3d := Subgroup( d12, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ] );
gap> Gs3d := SubgroupoidByPieces( Gd12, [ [[-36,-35,-34], s3d] ] );;
gap> SetName( s3d, "s3d" );  SetName( Gs3d, "Gs3d" );
gap> gend8 := GeneratorsOfGroup( d8 );;
gap> imhd8 := [ ( ), (15,20)(16,19)(17,18) ];;
gap> hd8 := GroupHomomorphismByImages( d8, s3d, gend8, imhd8 );
gap> homd8 := GroupoidHomomorphism( Gd8, Gs3d, hd8, [-34,-35,-36] );
groupoid homomorphism : Gd8 -> Gs3d 
gap> IsBijectiveOnObjects( homd8 );
true
gap> Display( homd8 );
groupoid mapping: [ Gd8 ] -> [ Gs3d ]
root homomorphism: [ [ (1,2,3,4), (1,3) ], [ (), (15,20)(16,19)(17,18) ] ]
images of objects: [ -34, -35, -36 ]
   images of rays: [ (), (), () ]

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