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#############################################################################
##
#W double.gd GAP4 package `Groupoids' Chris Wensley
##
##
## This file contains the declarations for basic double groupoids
##
#############################################################################
#C IsDoubleGroupoid( <ddwo> ) . . . . . . . . . . . . . . . and all inverses
#C IsBasicDoubleGroupoid( <ddwo> . . . . . . . . . . . . . . . . no prexmod
#V IsDoubleGroupoidFamily . . . . . . . . . . . family for double groupoids
#T IsDoubleGroupoidType( <gpd> )
#T IsDoubleGroupoidPiecesType( <dwo> )
#A GroupoidOfDoubleGroupoid( <dwo> )
##
DeclareCategory( "IsDoubleGroupoid",
IsMonoidWithObjects and IsDoubleGroupoidElementCollection );
DeclareCategory( "IsBasicDoubleGroupoid", IsDoubleGroupoid );
DeclareCategoryCollections( "IsDoubleGroupoid" );
IsDoubleGroupoidFamily := CollectionsFamily( IsDoubleGroupoidElementFamily );
BindGlobal( "IsDoubleGroupoidType",
NewType( IsDoubleGroupoidFamily,
IsMWOSinglePieceRep and IsDoubleGroupoid ) );
BindGlobal( "IsDoubleGroupoidPiecesType",
NewType( IsDoubleGroupoidFamily,
IsPiecesRep and IsDoubleGroupoid and IsAssociative ) );
DeclareAttribute( "GroupoidOfDoubleGroupoid", IsDoubleGroupoid );
#############################################################################
##
#O SquareOfArrows( <dmwo>, <elt>, <up>, <left>, <right>, <down> )
#O MultiplicativeSquareWithObjects( <dmwo>, <elt>, <up>, <lt>, <rt>, <dn> )
#O SquareOfArrowsNC( <dmwo>, <elt>, <up>, <left>, <right>, <down> )
##
DeclareOperation( "SquareOfArrows",
[ IsDoubleGroupoid, IsMultiplicativeElement,
IsObject, IsObject, IsObject, IsObject ] );
DeclareSynonym( "MultiplicativeSquareWithObjects", SquareOfArrows );
DeclareOperation( "SquareOfArrowsNC",
[ IsDoubleGroupoid, IsMultiplicativeElement,
IsObject, IsObject, IsObject, IsObject ] );
#############################################################################
##
#O ElementOfSquare( <swo> )
#O DownArrow( <swo> )
#O LeftArrow( <swo> )
#O UpArrow( <swo> )
#O RightArrow( <swo> )
#O BoundaryOfSquare( <swo> )
#O DoubleGroupoidOfSquare( <swo> )
##
DeclareOperation( "ElementOfSquare", [ IsDoubleGroupoidElement ] );
DeclareOperation( "DownArrow", [ IsDoubleGroupoidElement ] );
DeclareOperation( "LeftArrow", [ IsDoubleGroupoidElement ] );
DeclareOperation( "UpArrow", [ IsDoubleGroupoidElement ] );
DeclareOperation( "RightArrow", [ IsDoubleGroupoidElement ] );
DeclareOperation( "BoundaryOfSquare", [ IsDoubleGroupoidElement ] );
DeclareOperation( "DoubleGroupoidOfSquare", [ IsDoubleGroupoidElement ] );
#############################################################################
##
#P IsCommutingSquare( <sq> )
##
DeclareProperty( "IsCommutingSquare", IsDoubleGroupoidElement );
#############################################################################
##
#O VerticalProduct( <sq1>, <sq2> )
#O HorizontalProduct( <sq1>, <sq2> )
#O VerticalIdentities( <sq> )
#O HorizontalIdentities( <sq> )
#O VerticalInverse( <sq> )
#O HorizontalInverse( <sq> )
##
DeclareOperation( "VerticalProduct",
[ IsDoubleGroupoidElement, IsDoubleGroupoidElement ] );
DeclareOperation( "HorizontalProduct",
[ IsDoubleGroupoidElement, IsDoubleGroupoidElement ] );
DeclareOperation( "VerticalIdentities", [ IsDoubleGroupoidElement ] );
DeclareOperation( "HorizontalIdentities", [ IsDoubleGroupoidElement ] );
DeclareOperation( "VerticalInverse", [ IsDoubleGroupoidElement ] );
DeclareOperation( "HorizontalInverse", [ IsDoubleGroupoidElement ] );
#############################################################################
##
#O TransposedSquare( <sq> )
#P IsClosedUnderTransposition( <sq> )
##
DeclareOperation( "TransposedSquare",
[ IsDoubleGroupoidElement ] );
DeclareProperty( "IsClosedUnderTransposition", IsDoubleGroupoidElement );
#############################################################################
##
#O SinglePieceBasicDoubleGroupoid( <gpd> )
#F DoubleGroupoid( <gpd> )
#O DoubleGroupoidWithTrivialGroup( <obs> )
#O DoubleGroupoidWithSingleObject( <gpd>, <ob> )
##
DeclareOperation( "SinglePieceBasicDoubleGroupoid", [ IsGroupoid ] );
DeclareGlobalFunction( "DoubleGroupoid" );
DeclareOperation( "DoubleGroupoidWithTrivialGroup", [ IsList ] );
DeclareOperation( "DoubleGroupoidWithSingleObject", [ IsGroup, IsObject ] );
#############################################################################
##
#O DoubleGroupoidHomomorphism( <src> <rng> <gpdhom> )
#C IsDoubleGroupoidHomomorphism( <map> )
#C IsDoubleGroupoidHomomorphismCollection . . category of colls of dgpd homs
#P IsDoubleGroupoidEndomorphism( <mor> )
#A UnderlyingGroupoidHomomorphism( <hom> )
##
DeclareOperation( "DoubleGroupoidHomomorphism",
[ IsDoubleGroupoid, IsDoubleGroupoid, IsGroupoidHomomorphism ] );
DeclareCategory( "IsDoubleGroupoidHomomorphism",
IsMagmaWithObjectsHomomorphism );
DeclareCategoryCollections( "IsDoubleGroupoidHomomorphism" );
DeclareSynonym( "IsDoubleGroupoidEndomorphism",
IsDoubleGroupoidHomomorphism and IsEndomorphismWithObjects );
DeclareAttribute( "UnderlyingGroupoidHomomorphism",
IsDoubleGroupoidHomomorphism );
##############################################################################
##
#R IsDefaultDoubleGroupoidHomomorphismRep( <map> )
##
## A mapping of connected double groupoids is determined by
## a groupoid homomorphism, and this defines:
## - mapping from the root group in the source to that in the range,
## - images for the objects,
## - images for the rays.
## In this representation we essentially duplicate the gpd hom rep.
##
DeclareRepresentation( "IsDefaultDoubleGroupoidHomomorphismRep",
IsDoubleGroupoidHomomorphism and IsAttributeStoringRep
and IsComponentObjectRep, [ "Source", "Range", "PiecesOfMapping",
"UnderlyingGroupoidHomomorphism" ] );
#############################################################################
##
#V DoubleGroupoidHomomorphismFamily . . family for homs of double groupoids
#T DoubleGroupoidHomomorphismType . . . . type for homs of double groupoids
##
BindGlobal( "DoubleGroupoidHomomorphismFamily",
NewFamily( "DoubleGroupoidHomomorphismFamily",
IsDoubleGroupoidHomomorphism,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "DoubleGroupoidHomomorphismType",
NewType( DoubleGroupoidHomomorphismFamily,
IsDefaultDoubleGroupoidHomomorphismRep
and IsDoubleGroupoidHomomorphism ) );
############################################################################
##
#E double.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##
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