GAP (XMod) - Chapter 8: Crossed modules of groupoids

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

8 Crossed modules of groupoids
8.1 Constructions for crossed modules of groupoids
8.1-1 PreXModWithObjectsObj

### 8 Crossed modules of groupoids

#### 8.1 Constructions for crossed modules of groupoids

A typical example of a crossed module over a groupoid has as range a connected groupoid which is a direct product of a group and a complete graph, and as source a totally disconnected groupoid, with the same objects. The boundary morphism is constant on objects. For details and other references see [AW10].

##### 8.1-1 PreXModWithObjectsObj

`‣ PreXModWithObjectsObj`( obs, bdy, act ) | ( operation )

`‣ DiscreteNormalPreXModWithObjects`( gpd, gp ) | ( operation )

The next stage of development of this package will be to implement constuctions of crossed modules over groupoids. This will require further developments in the Gpd package. The following example shows what has been achieved in an earlier version, but which fails in GAP 4.7.

    gap> d8 := Group( (1,2,3,4), (1,3) );;
    gap> SetName( d8, "d8" );
    gap> Gd8 := SinglePieceGroupoid( d8, [-9,-8,-7] );;
    gap> Display( Gd8 );
    single piece groupoid:
      objects: [ -9, -8, -7 ]
        group: d8 = <[ (1,2,3,4), (1,3) ]>
    gap> k4 := Subgroup( d8, [ (1,2)(3,4), (1,3)(2,4) ] );;
    gap> PX0 := DiscreteNormalPreXModWithObjects( Gd8, k4 );
    homogeneous, discrete groupoid with:
      group: Group( [ (1,2)(3,4), (1,3)(2,4) ] ) >
    objects: [ -9, -8, -7 ]
    #I  now need to be able to test:   ok := IsXMod( PM );
    <magma>
    gap> Source(PX0);
    perm homogeneous, discrete groupoid: < Group( [ (1,2)(3,4), (1,3)(2,4) ] ),
    [ -9, -8, -7 ] >

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 Bib Ind

generated by GAPDoc2HTML