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#############################################################################
##
#W  double.gi                 GAP4 package `groupoids'          Chris Wensley
##
##  This file contains the implementations for basic double groupoids
##

#####################  MULT SQUARE WITH OBJECTS  ###########################

############################################################################
##
#M  SquareOfArrows( <dmwo>, <elt>, <down>, <left>, <up>, <right> )
#M  SquareOfArrowsNC( <dmwo>, <elt>, <down>, <left>, <up>, <right> )
##
InstallMethod( SquareOfArrowsNC,
    "for double groupoid, element, up, left, right and down", true,
    [ IsDoubleGroupoid, IsMultiplicativeElement,
      IsObject, IsObject, IsObject, IsObject ], 0,
function( dg, m, u, l, r, d )

    local elt, fam;

    fam := IsGroupoidElementFamily;
    elt := Objectify( IsDoubleGroupoidElementType, [ dg, m, u, l, r, d ] );
    return elt;
end );

InstallMethod( SquareOfArrows,
    "for double groupoid with objects, element, up, left, right and down",
    true, [ IsBasicDoubleGroupoid, IsMultiplicativeElement,
            IsObject, IsObject, IsObject, IsObject ], 0,
function( dmwo, e, u, l, r, d )

    local  gpd, ok, piece, loop, sq;

    gpd := dmwo!.groupoid;
    ok := ForAll( [u,l,r,d], x -> x in gpd )
             and ( TailOfArrow( u ) = TailOfArrow( l ) )
             and ( HeadOfArrow( u ) = TailOfArrow( r ) )
             and ( HeadOfArrow( l ) = TailOfArrow( d ) )
             and ( HeadOfArrow( r ) = HeadOfArrow( d ) );
    if not ok then
        Info( InfoGroupoids, 1, "the four arrows do not form a square" );
        return fail;
    else
        if IsSinglePiece( gpd ) then
            piece := dmwo;
        else
            piece := PieceOfObject( gpd, TailOfArrow( d ) );
        fi;
        loop := d^-1 * l^-1 * u * r;
        ok := ( e = loop![2] );
        if not ok then
            Info( InfoGroupoids, 2, "here" );
            Info( InfoGroupoids, 1, "element ", e,
                               " <> boundary element ", loop![2] );
            return fail;
        fi;
        sq := SquareOfArrowsNC( dmwo, e, u, l, r, d );
        return sq;
    fi;
end );

InstallOtherMethod( SquareOfArrows,
    "for double groupoid with objects, up, left, right and down",
    true, [ IsDoubleGroupoid, IsObject, IsObject, IsObject, IsObject ], 0,
function( dmwo, u, l, r, d )

    local  gpd, ok, piece, loop, m, sq;

    gpd := dmwo!.groupoid;
    ok := ForAll( [u,l,r,d], x -> x in gpd )
             and ( TailOfArrow( u ) = TailOfArrow( l ) )
             and ( HeadOfArrow( u ) = TailOfArrow( r ) )
             and ( HeadOfArrow( l ) = TailOfArrow( d ) )
             and ( HeadOfArrow( r ) = HeadOfArrow( d ) );
    if not ok then
        Info( InfoGroupoids, 1, "the four arrows do not form a square" );
        return fail;
    else
        if IsSinglePiece( gpd ) then
            piece := dmwo;
        else
            piece := PieceOfObject( gpd, TailOfArrow( d ) );
        fi;
        m := ( d^-1 * l^-1 * u * r )![2];
        sq := SquareOfArrowsNC( dmwo, m, u, l, r, d );
        return sq;
    fi;
end );

#############################################################################
##
#M  ElementOfSquare
#M  DownArrow
#M  LeftArrow
#M  UpArrow
#M  DownArrow
#M  BoundaryOfSquare
##
InstallMethod( DoubleGroupoidOfSquare, "generic method for double gpd element",
    true, [ IsDoubleGroupoidElement ], 0, s -> s![1] );

InstallMethod( ElementOfSquare, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0, s -> s![2] );

InstallMethod( UpArrow, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0, s -> s![3] );

InstallMethod( LeftArrow, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0, e -> e![4] );

InstallMethod( RightArrow, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0, s -> s![5] );

InstallMethod( DownArrow, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0, s -> s![6] );

InstallMethod( BoundaryOfSquare, "generic method for double groupoid element",
    true, [ IsDoubleGroupoidElement ], 0,
    s -> s![6]^(-1) * s![4]^(-1) * s![3] * s![5] );

#############################################################################
##
#M  String, ViewString, PrintString, ViewObj, PrintObj
##  . . . . . . . . . . . . . . . . . . for elements in a double groupoid
##
InstallMethod( String, "for a square in a double groupoid", true,
    [ IsMultiplicativeElementWithObjects ], 0,
function( e )
    return( STRINGIFY( "[", String( e![2] ), " : ", String( e![3] ),
                       " -> ", String( e![4] ), "]" ) );
end );

InstallMethod( ViewString, "for a square in a double groupoid", true,
    [ IsDoubleGroupoidElement ], 0, String );

InstallMethod( PrintString, "for a square in a double groupoid", true,
    [ IsDoubleGroupoidElement ], 0, String );

InstallMethod( ViewObj, "for a square in a double groupoid", true,
    [ IsDoubleGroupoidElement ], 0, PrintObj );

InstallMethod( PrintObj, "for a square in a double groupoid",
    [ IsDoubleGroupoidElement ],
function ( sq )
    local up, lt, rt, dn, es, spaces, dashes,
          upstr, ltstr, rtstr, dnstr, uplen, ltlen, rtlen, dnlen, esstr, eslen,
          toplen, midlen, botlen, maxlen, topgap, midgap, botgap;
    up := UpArrow( sq );
    lt := LeftArrow( sq );
    rt := RightArrow( sq );
    dn := DownArrow( sq );
    es := ElementOfSquare( sq );
    spaces := "                                                  ";
    dashes := "--------------------------------------------------";
    upstr := [ String( up![2] ), String( up![3] ), String( up![4] ) ];
    uplen := List( upstr, s -> Length(s) );
    ltstr := [ String( lt![2] ), String( lt![3] ), String( lt![4] ) ];
    ltlen := List( ltstr, s -> Length(s) );
    rtstr := [ String( rt![2] ), String( rt![3] ), String( rt![4] ) ];
    rtlen := List( rtstr, s -> Length(s) );
    dnstr := [ String( dn![2] ), String( dn![3] ), String( dn![4] ) ];
    dnlen := List( dnstr, s -> Length(s) );
    esstr := String( es );
    eslen := Length( esstr );
    toplen := uplen[2]+2 + 6 + uplen[1] + 6 + uplen[3]+2;
    midlen := ltlen[1] + 6 + eslen + 6 + rtlen[1];
    botlen := dnlen[2]+2 + 6 + dnlen[1] + 6 + dnlen[3]+2;
    maxlen := Maximum( [ toplen, midlen, botlen ] );
    topgap := 4;
    midgap := 8;
    botgap := 4;
    if ( maxlen = midlen ) then
        topgap := 4 + Int( (maxlen - toplen)/2 );
        botgap := 4 + Int( (maxlen - botlen)/2 );
    elif ( maxlen = botlen ) then
        topgap := topgap + Int( (botlen-toplen)/2 );
        midgap := midgap - 2 + Int( (botlen-midlen)/2 );
    elif ( maxlen = toplen ) then
        botgap := botgap + Int( (toplen-botlen)/2 );
        midgap := midgap - 2 + Int( (toplen-midlen)/2 );
    fi;
    Print( "[", up![3], "] ", dashes{[1..topgap]}, " ", up![2],
           " ", dashes{[1..topgap-1]}, "> [", up![4], "]\n" );
    Print( "  |  ", spaces{[1..maxlen-11]}, "   |\n" );
    Print( lt![2], spaces{[1..midgap]}, es, spaces{[1..midgap]}, rt![2], "\n" );
    Print( "  V  ", spaces{[1..maxlen-11]}, "   V\n" );
    Print( "[", dn![3], "] ", dashes{[1..botgap]}, " ", dn![2],
           " ", dashes{[1..botgap-1]}, "> [", dn![4], "]" );
end );

InstallMethod( ViewObj, "for an element in a magma with objects",
    [ IsMultiplicativeElementWithObjects ], PrintObj );

#############################################################################
##
#M  IsCommutingSquare
##
InstallMethod( IsCommutingSquare, "for a square in a double groupoid", true,
    [ IsDoubleGroupoidElement ], 0,
function( sq )
    return sq![4] * sq![6] = sq![3] * sq![5];
end );

#############################################################################
##
#M  HorizontalProduct( s1, s2 )
##      . . . . horizantal composition of squares in a basic double groupoid
##
InstallMethod( HorizontalProduct,
    "for two squares in a basic double groupoid", true,
    [ IsDoubleGroupoidElement, IsDoubleGroupoidElement ], 0,
function( s1, s2 )

    local m;

    ## elements are composable?
    if not ( s1![5] = s2![4] ) then
        Info( InfoGroupoids, 1, "right arrow of s1 <> left arrow of s2" );
        return fail;
    fi;
    if not ( s1![1] = s2![1] ) then
        Info( InfoGroupoids, 1, "s1 and s2 are in different double groupoids" );
        return fail;
    fi;
    m := s1![2]^s2![6]![2] * s2![2];
    return SquareOfArrowsNC( s1![1], m, s1![3]*s2![3], s1![4],
                                        s2![5], s1![6]*s2![6] );
end );

#############################################################################
##
#M  VerticalProduct( s1, s2 )
##  . . . . . . . vertical composition of squares in a basic double groupoid
##
InstallMethod( VerticalProduct, "for two squares in a basic double groupoid",
    true, [ IsDoubleGroupoidElement, IsDoubleGroupoidElement ], 0,
function( s1, s2 )

    local m;

    ## elements are composable?
    if not ( s1![6] = s2![3] ) then
        Info( InfoGroupoids, 1, "down arrow of s1 <> up arrow of s2" );
        return fail;
    fi;
    if not ( s1![1] = s2![1] ) then
        Info( InfoGroupoids, 1, "s1 and s2 are in different double groupoids" );
        return fail;
    fi;
    m := s2![2] * s1![2]^s2![5]![2];
    return SquareOfArrowsNC( s1![1], m, s1![3], s1![4]*s2![4],
                                        s1![5]*s2![5], s2![6] );
end );

#############################################################################
##
#M  HorizontalIdentities( s )
##
InstallMethod( HorizontalIdentities, "for a square in a basic double groupoid",
    true, [ IsDoubleGroupoidElement ], 0,
function( s )

    local b, c, gpd, u, v, w, x, one, ui, vi, wi, xi, lti, rti;

    b := LeftArrow( s );
    c := RightArrow( s );
    gpd := (s![1])!.groupoid;
    u := b![3];  w := b![4];  v := c![3];  x := c![4];
    one := One( b![2] );
    ui := Arrow( gpd, one, u, u );
    vi := Arrow( gpd, one, v, v );
    wi := Arrow( gpd, one, w, w );
    xi := Arrow( gpd, one, x, x );
    lti := SquareOfArrowsNC( s![1], one, ui, b, b, wi );
    rti := SquareOfArrowsNC( s![1], one, vi, c, c, xi );
    return [ lti, rti ];
end );

#############################################################################
##
#M  VerticalIdentities( s )
##
InstallMethod( VerticalIdentities, "for a square in a basic double groupoid",
    true, [ IsDoubleGroupoidElement ], 0,
function( s )

    local a, d, gpd, u, v, w, x, one, ui, vi, wi, xi, upi, dni;

    a := UpArrow( s );
    d := DownArrow( s );
    gpd := (s![1])!.groupoid;
    u := a![3];  v := a![4];  w := d![3];  x := d![4];
    one := One( a![2] );
    ui := Arrow( gpd, one, u, u );
    vi := Arrow( gpd, one, v, v );
    wi := Arrow( gpd, one, w, w );
    xi := Arrow( gpd, one, x, x );
    upi := SquareOfArrowsNC( s![1], one, a, ui, vi, a );
    dni := SquareOfArrowsNC( s![1], one, d, wi, xi, d );
    return [ upi, dni ];
end );

#############################################################################
##
#M  HorizontalInverse( s )
##
InstallMethod( HorizontalInverse, "for a square in a basic double groupoid",
    true, [ IsDoubleGroupoidElement ], 0,
function( s )

    local a, b, c, d, gpd, bdy;

    a := UpArrow( s );
    b := LeftArrow( s );
    c := RightArrow( s );
    d := DownArrow( s );
    gpd := (s![1])!.groupoid;
    bdy := d * c^-1 * a^-1 * b;
    return SquareOfArrowsNC( s![1], bdy![2], a^-1, c, b, d^-1 );
end );

#############################################################################
##
#M  VerticalInverse( s )
##
InstallMethod( VerticalInverse, "for a square in a basic double groupoid",
    true, [ IsDoubleGroupoidElement ], 0,
function( s )

    local a, b, c, d, gpd, bdy;

    a := UpArrow( s );
    b := LeftArrow( s );
    c := RightArrow( s );
    d := DownArrow( s );
    gpd := (s![1])!.groupoid;
    bdy := a^-1 * b * d * c^-1;
    return SquareOfArrowsNC( s![1], bdy![2], d, b^-1, c^-1, a );
end );

#############################################################################
##
#M  TransposedSquare( sq )
##
InstallMethod( TransposedSquare, "for a square in a double groupoid", true,
    [ IsDoubleGroupoidElement ], 0,
function( sq )
    local dg, a, b, c, d, m, tsq;

    dg := DoubleGroupoidOfSquare( sq );
    a := UpArrow( sq );
    b := LeftArrow( sq );
    c := RightArrow( sq );
    d := DownArrow( sq );
    m := BoundaryOfSquare( sq )![2];
    return SquareOfArrowsNC( dg, m^-1, b, a, d, c );
end );

InstallMethod( IsClosedUnderTransposition, "for a square in a double groupoid",
    true, [ IsDoubleGroupoidElement ], 0,
function( sq )
    local tsq;

    tsq := TransposedSquare( sq );
    return sq = tsq;
end );

#############################################################################
##
#M  \in( <sq>, <dg> ) . . . . . . . test if a square is in a double groupoid
##

InstallMethod( \in, "for a square in a standard double groupoid", true,
    [ IsDoubleGroupoidElement, IsDoubleGroupoid and IsSinglePiece ], 0,
function( sq, dg )
    return ( sq![1] = dg );
end );

InstallMethod( \in, "for a square in a double groupoid with pieces", true,
    [ IsDoubleGroupoidElement, IsDoubleGroupoid and HasPieces ], 0,
function( sq, dg )
    local u, p;

    u := sq![3]![3];
    p := PieceOfObject( dg, u );
    if p = fail then
        return false;
    else
        return sq in p;
    fi;
end );

#############################################################################
##
#M  \=( <dg1>, <dg2> ) . . . . . . . . . . test equality of double groupoids
##

InstallMethod( \=, "for two double groupoids", true,
    [ IsDoubleGroupoid, IsDoubleGroupoid ], 0,
function( dg1, dg2 )

    local p1, p2, i;

    if ( IsSinglePiece( dg1 ) and IsSinglePiece( dg2 ) ) then
        return ( ( dg1!.objects = dg2!.objects )
             and ( dg1!.groupoid = dg2!.groupoid ) );
    elif ( IsSinglePiece( dg1 ) or IsSinglePiece( dg2 ) ) then
        return false;
    else
        p1 := Pieces( dg1 );
        p2 := Pieces( dg2 );
        if not ( Length( p1 ) = Length( p2 ) ) then
            return false;
        else
            for i in [1..Length( p1 )] do
                if ( p1[i] <> p2[i] ) then
                    return false;
                fi;
            od;
            return true;
        fi;
    fi;
end );

#############################################################################
##
#M  Size
##
InstallOtherMethod( Size, "generic method for a double groupoid", true,
    [ IsDoubleGroupoid ], 0,
function( dg )

    local gpd, gp, obs, p, s;

    gpd := dg!.groupoid;
    gp := gpd!.magma;
    obs := dg!.objects;
    if IsBasicDoubleGroupoid( dg ) then
        return Size( gp )^4 * Length( obs )^4;
    else
        TryNextMethod();
    fi;
end );

#############################################################################
##
#M  String, ViewString, PrintString, ViewObj, PrintObj
##  . . . . . . . . . . . . . . . . . . . . . . . . . . for a double groupoid
##
InstallMethod( String, "for a double groupoid", true, [ IsDoubleGroupoid ], 0,
function( mwo )
    return( STRINGIFY( "groupoid" ) );
end );

InstallMethod( ViewString, "for a double groupoid", true,
    [ IsDoubleGroupoid ], 0, String );

InstallMethod( PrintString, "for a double groupoid", true,
    [ IsDoubleGroupoid ], 0, String );

InstallMethod( ViewObj, "for a double groupoid", true,
    [ IsDoubleGroupoid ], 0, PrintObj );

InstallMethod( ViewObj, "for a double groupoid", true,
    [ IsDoubleGroupoid and IsSinglePiece ], 0,
function( dgpd )
    Print( "single piece double groupoid with:\n" );
    Print( " groupoid = ", dgpd!.groupoid, "\n" );
    Print( "    group = ", dgpd!.groupoid!.magma, "\n" );
    Print( "  objects = ", dgpd!.objects );
end );

InstallMethod( PrintObj, "for a double groupoid", true,
    [ IsDoubleGroupoid and IsSinglePiece ], 0,
function( dgpd )
    Print( "single piece double groupoid with:\n" );
    Print( " groupoid = ", dgpd!.groupoid, "\n" );
    Print( "    group = ", dgpd!.groupoid!.magma, "\n" );
    Print( "  objects = ", dgpd!.objects, "\n" );
end );

InstallMethod( ViewObj, "for more than one piece", true,
    [ IsDoubleGroupoid and IsPiecesRep ], 10,
function( ddwo )

    local i, pieces, np;

    pieces := Pieces( ddwo );
    np := Length( pieces );
    Print( "double groupoid having ", np, " pieces :-\n" );
    for i in [1..np] do
        Print( i, ": ", pieces[i] );
        if HasName( pieces[i] ) then
            Print( "\n" );
        fi;
    od;
end );

InstallMethod( PrintObj, "for more than one piece", true,
    [ IsDoubleGroupoid and IsPiecesRep ], 0,
function( dg )

    local i, pieces, np;

    pieces := Pieces( dg );
    np := Length( pieces );
    Print( "domain with objects having ", np, " pieces :-\n" );
    for i in [1..np] do
        Print( pieces[i] );
        if HasName( pieces[i] ) then
            Print( "\n" );
        fi;
    od;
end );

#############################################################################
##
#M  Display( <dg> ) . . . . . . . . . . . . . display a magma with objects
##
InstallMethod( Display, "for a double groupoid", true, [ IsDoubleGroupoid ], 0,
function( dg )

    local comp, c, i, m, len;

    if IsSinglePiece( dg ) then
        if IsDirectProductWithCompleteDigraph( dg ) then
            Print( "Single constituent magma with objects: " );
            if HasName( dg ) then
                Print( dg );
            fi;
            Print( "\n" );
            Print( "  objects: ", dg!.objects, "\n" );
            m := dg!.magma;
            Print( "    magma: " );
            if HasName( m ) then
                Print( m, " = <", GeneratorsOfMagma( m ), ">\n" );
            else
                Print( m, "\n" );
            fi;
        else
            TryNextMethod();
        fi;
    else
        comp := Pieces( dg );
        len := Length( comp );
        Print( "Magma with objects with ", len, " pieces:\n" );
        for i in [1..len] do
            c := comp[i];
            if IsDirectProductWithCompleteDigraph( c ) then
                Print( "< objects: ", c!.objects, "\n" );
                m := c!.magma;
                Print( "    magma: " );
                if HasName( m ) then
                    Print( m, " = <", GeneratorsOfMagma( m ), "> >\n" );
                else
                    Print( m, " >\n" );
                fi;
            else
                TryNextMethod();
            fi;
        od;
    fi;
end );

###################  DOUBLE DOMAIN WITH OBJECTS   ##########################

#############################################################################
##
#F  DoubleGroupoid( <pieces> )          double groupoid as list of pieces
#F  DoubleGroupoid( <gpd> )             basic single piece double groupoid
#F  DoubleGroupoid( <gpd> )             when the groupoid has pieces
#O  DoubleGroupoidWithTrivialGroup( <obs> )
##
InstallGlobalFunction( DoubleGroupoid, function( arg )

    local nargs, gpd, pieces, len, L, dg;

    nargs := Length( arg );
    # list of pieces
    if ( ( nargs = 1 ) and IsList( arg[1] ) ) then
        if ForAll( arg[1], IsDoubleGroupoid ) then
            Info( InfoGroupoids, 2, "ByUnion" );
            return UnionOfPieces( arg[1] );
        elif ForAll( arg[1], IsGroupoid ) then
            pieces := arg[1];
            len := Length( pieces );
            if ( len = 1 ) then
                Info( InfoGroupoids, 2, "SinglePieceBasicDoubleGroupoid" );
                return SinglePieceBasicDoubleGroupoid( gpd );
            else
                L := List( pieces, p -> SinglePieceBasicDoubleGroupoid( p ) );
                dg := UnionOfPieces( L );
                return dg;
            fi;
        fi;
    #  groupoid
    elif ( ( nargs = 1 ) and IsGroupoid( arg[1] ) ) then
        gpd := arg[1];
        pieces := Pieces( gpd );
        len := Length( pieces );
        if ( len = 1 ) then
            Info( InfoGroupoids, 2, "SinglePieceBasicDoubleGroupoid" );
            return SinglePieceBasicDoubleGroupoid( gpd );
        else
            L := List( pieces, p -> SinglePieceBasicDoubleGroupoid( p ) );
            dg := UnionOfPieces( L );
            return dg;
        fi;
    else
        return fail;
    fi;
end );

InstallMethod( SinglePieceBasicDoubleGroupoid, "for a groupoid", true,
    [ IsGroupoid ], 0,
function( gpd )

    local dgpd;

    dgpd := rec( groupoid := gpd, objects := ObjectList( gpd ) );
    ObjectifyWithAttributes( dgpd, IsDoubleGroupoidType,
        IsSinglePiece, true,
        IsAssociative, true,
        IsCommutative, IsCommutative( gpd!.magma ),
        IsBasicDoubleGroupoid, true,
        GroupoidOfDoubleGroupoid, gpd );
    return dgpd;
end );

InstallMethod( DoubleGroupoidWithTrivialGroup, "for a list of objects",
    true, [ IsList ], 0,
function( obs )

    local gp, gpd, dg;

    gp := Group( () );
    gpd := SinglePieceGroupoid( gp, obs );
    dg := SinglePieceBasicDoubleGroupoid( gpd );
    return dg;
end );

InstallMethod( DoubleGroupoidWithSingleObject, "for a group and an object",
    true, [ IsGroup, IsObject ], 0,
function( gp, obj )

    local gpd, dg;

    gpd := MagmaWithSingleObject( gp, obj );
    dg := SinglePieceBasicDoubleGroupoid( gpd );
    return dg;
end );

############################################################################
##
#M  GroupoidOfDoubleGroupoid( <src> <rng> <hom> )
##
InstallMethod( GroupoidOfDoubleGroupoid, "for a double groupoid", true,
    [ IsDoubleGroupoid ], 0,
function( dg )
    local pieces, gpds;
    if ( HasIsSinglePiece( dg ) and IsSinglePiece( dg ) ) then
        return dg!.groupoid;
    else
        pieces := Pieces( dg );
        gpds := List( pieces, p -> p!.groupoid );
        return UnionOfPieces( gpds );
    fi;
end );

############################################################################
##
#M  DoubleGroupoidHomomorphism( <src> <rng> <hom> ) . . from a groupoid hom
##
InstallMethod( DoubleGroupoidHomomorphism, "for a groupoid homomorphism",
    true, [ IsDoubleGroupoid, IsDoubleGroupoid, IsGroupoidHomomorphism ], 0,
function( src, rng, hom )

    local sgpd, rgpd, map;

    sgpd := src!.groupoid;
    rgpd := rng!.groupoid;
    if not ( sgpd = Source( hom ) ) and ( rgpd = Range( hom ) ) then
        Error( "hom not a map between the underlying groupoids" );
    fi;
    map := rec();
    ObjectifyWithAttributes( map, DoubleGroupoidHomomorphismType,
        Source, src,
        Range, rng,
        UnderlyingGroupoidHomomorphism, hom,
        IsGeneralMappingWithObjects, true,
        IsGroupWithObjectsHomomorphism, true,
        RespectsMultiplication, true );
    return map;
end );

InstallMethod( PrintObj, "for a double groupoid homomorphism", true,
    [ IsDoubleGroupoidHomomorphism ], 0,
function ( hom )

    local h;

    if HasPiecesOfMapping( hom ) then
        Print( "double groupoid homomorphism from several pieces : \n" );
        for h in PiecesOfMapping( hom ) do
            Print( h, "\n" );
        od;
    else
        Print( "double groupoid homomorphism : " );
        if ( HasName( Source(hom) ) and HasName( Range(hom) ) ) then
            Print( Source(hom), " -> ", Range(hom), "\n" );
        else
            Print( "based on ", UnderlyingGroupoidHomomorphism( hom ), "\n" );
        fi;
    fi;
end );

############################################################################
##
#M  Display
##
InstallMethod( Display, "for double groupoid homomorphism", true,
    [ IsDoubleGroupoidHomomorphism ], 0,
function ( map )
    Print( "double groupoid homomorphism: " );
    Print( "[ ", Source(map), " ] -> [ ", Range(map), " ]\n" );
    Print( "with underlying groupoid homomorphism:\n" );
    Display( UnderlyingGroupoidHomomorphism( map ) );
end );

############################################################################
##
#M  ImageElm
##
InstallOtherMethod( ImageElm, "for a double groupoid homomorphism and a square",
    true, [ IsDoubleGroupoidHomomorphism, IsDoubleGroupoidElement ], 0,
function ( hom, sq )

    local mor, up, lt, rt, dn, e, isq;

    if not ( sq in Source( hom ) ) then
        Error( "the square sq is not in the source of hom" );
    fi;
    mor := UnderlyingGroupoidHomomorphism( hom );
    up := ImageElm( mor, UpArrow( sq ) );
    lt := ImageElm( mor, LeftArrow( sq ) );
    rt := ImageElm( mor, RightArrow( sq ) );
    dn := ImageElm( mor, DownArrow( sq ) );
    e := dn![2]^-1 * lt![2]^-1 * up![2] * rt![2];
    isq := SquareOfArrowsNC( Range( hom ), e, up, lt, rt, dn );
    return isq;
end );

############################################################################
##
#E double.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##

Quelle double.gi Sprache: unbekannt

[ Dauer der Verarbeitung: 0.29 Sekunden (vorverarbeitet) ]