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#############################################################################
##
#W  double.gi                   GAP4 package `XMod'             Chris Wensley
##
#Y  Copyright (C) 2000-2025, Chris Wensley,
##
##  This file contains the implementations for double groupoids
##

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

InstallMethod( SinglePieceDoubleGroupoid,
    "for groupoid and precrossed module", true, [ IsGroupoid, IsPreXMod ], 0,
function( gpd, px )

    local dgpd;

    dgpd := rec( groupoid := gpd, prexmod := px,
                 objects := ObjectList( gpd ) );
    ObjectifyWithAttributes( dgpd, IsDoubleGroupoidType,
        IsSinglePiece, true,
        IsAssociative, true,
        IsCommutative, IsCommutative( gpd!.magma )
                       and IsCommutative( Source( px ) ),
        IsDoubleGroupoidWithPreXMod, true );
    return dgpd;
end );

InstallMethod( EnhancedBasicDoubleGroupoid, "for a basic double groupoid",
    true, [ IsBasicDoubleGroupoid ], 0,
function( bdg )

    local gpd, obs, gp, px, dgpd;

    if not IsSinglePiece( bdg ) then
        Error( "only working with single pieces so far" );
    fi;
    gpd := bdg!.groupoid;
    obs := bdg!.objects;
    gp := RootGroup( gpd );
    px := XModByNormalSubgroup( gp, gp );
    dgpd := SinglePieceDoubleGroupoid( gpd, px );
    return dgpd;
end );

InstallMethod( DoubleGroupoidWithZeroBoundary, "for a groupoid and a group",
    true, [ IsGroupoid, IsGroup ], 0,
function( gpd, src )

    local gp, px, dgpd;

    if not IsSinglePiece( gpd ) then
        Error( "only working with single pieces so far" );
    fi;
    gp := RootGroup( gpd );
    px := PreXModWithTrivialRange( src, gp );
    dgpd := SinglePieceDoubleGroupoid( gpd, px );
    return dgpd;
end );

InstallMethod( ViewObj, "for a double groupoid with prexmod", true,
    [ IsDoubleGroupoidWithPreXMod and IsSinglePiece ], 0,
function( dgpd )
    local pxm, isxmod;
    pxm := dgpd!.prexmod;
    isxmod := HasIsXMod( pxm ) and IsXMod( pxm );
    Print( "single piece double groupoid with:\n" );
    if isxmod then
        Print( "     xmod = ", pxm, "\n" );
    else
        Print( "  prexmod = ", pxm, "\n" );
    fi;
    Print( " groupoid = ", dgpd!.groupoid, "\n" );
    Print( "    group = ", dgpd!.groupoid!.magma, "\n" );
    Print( "  objects = ", dgpd!.objects );
end );

InstallMethod( PrintObj, "for a double groupoid with prexmod", true,
    [ IsDoubleGroupoidWithPreXMod and IsSinglePiece ], 0,
function( dgpd )
    local pxm, isxmod;
    pxm := dgpd!.prexmod;
    isxmod := HasIsXMod( pxm ) and IsXMod( pxm );
    Print( "single piece double groupoid with:\n" );
    if isxmod then
        Print( "     xmod = ", pxm, "\n" );
    else
        Print( "  prexmod = ", pxm, "\n" );
    fi;
    Print( " groupoid = ", dgpd!.groupoid, "\n" );
    Print( "    group = ", dgpd!.groupoid!.magma, "\n" );
    Print( "  objects = ", dgpd!.objects, "\n" );
end );

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

    local  gpd, px, bdy, ime, ok, piece, loop, sq;

    gpd := dgpd!.groupoid;
    px := dgpd!.prexmod;
    bdy := Boundary( px );
    ime := ImageElm( bdy, e );
    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( InfoXMod, 1, "the four arrows do not form a square" );
        return fail;
    else
        if IsSinglePiece( gpd ) then
            piece := dgpd;
        else
            piece := PieceOfObject( gpd, TailOfArrow( d ) );
        fi;
        loop := d^-1 * l^-1 * u * r;
        ok := ( ime = loop![2] );
        if not ok then
            Info( InfoXMod, 1, "element ", e, " has image ", ime,
                               " <> boundary element ", loop![2] );
            return fail;
        fi;
        sq := SquareOfArrowsNC( dgpd, e, u, l, r, d );
        return sq;
    fi;
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 dgpd, m, px, act, aut;

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

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

    local dgpd, m, px, act, aut;

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

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

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