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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:

- a homomorphism `hom` from the group of G to the group of H.

- a list `imobs` of the images of the objects of G;

    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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