| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/groupoids/lib/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 11.8.2025 mit Größe 6 kB |
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Quelle double.gd Sprache: unbekannt |
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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 ## [ Dauer der Verarbeitung: 0.21 Sekunden (vorverarbeitet) ] |
2026-03-28