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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
## CAP package
##
## Copyright 2014, Sebastian Gutsche, TU Kaiserslautern
## Sebastian Posur, RWTH Aachen
##
#! @Chapter Universal Objects
##
#############################################################################
####################################
##
## Kernel
##
####################################
####################################
## Functorial operations
####################################
##
InstallMethod( KernelObjectFunctorial,
[ IsList ],
function( morphism_of_morphisms )
return KernelObjectFunctorialWithGivenKernelObjects(
KernelObject( morphism_of_morphisms[1] ),
morphism_of_morphisms[1], morphism_of_morphisms[2][1], morphism_of_morphisms[3],
KernelObject( morphism_of_morphisms[3] ) );
end );
##
InstallMethod( KernelObjectFunctorial,
[ IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( alpha, mu, alpha_prime )
return KernelObjectFunctorialWithGivenKernelObjects( KernelObject( alpha ), alpha, mu, alpha_prime, KernelObject( alpha_prime ) );
end );
####################################
##
## Cokernel
##
####################################
####################################
## Functorial operations
####################################
##
InstallMethod( CokernelObjectFunctorial,
[ IsList ],
function( morphism_of_morphisms )
return CokernelObjectFunctorialWithGivenCokernelObjects(
CokernelObject( morphism_of_morphisms[1] ),
morphism_of_morphisms[1], morphism_of_morphisms[2][2], morphism_of_morphisms[3],
CokernelObject( morphism_of_morphisms[3] ) );
end );
##
InstallMethod( CokernelObjectFunctorial,
[ IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( alpha, nu, alpha_prime )
return CokernelObjectFunctorialWithGivenCokernelObjects( CokernelObject( alpha ), alpha, nu, alpha_prime, CokernelObject( alpha_prime ) );
end );
####################################
##
## Coproduct and Pushout
##
####################################
####################################
##
## Coproduct
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( Coproduct,
[ IsList ],
function( diagram )
return CoproductOp( diagram, diagram[1] );
end );
##
InstallMethod( Coproduct,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return CoproductOp( [ object_1, object_2 ], object_1 );
end );
##
InstallMethod( Coproduct,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
return CoproductOp( [ object_1, object_2, object_3 ], object_1 );
end );
##
InstallGlobalFunction( UniversalMorphismFromCoproduct,
function( arg )
local diagram;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismFromCoproductOp( arg[1], arg[2], arg[1][1] );
elif Length( arg ) = 1 and IsList( arg[ 1 ] ) then
arg := arg[ 1 ];
fi;
diagram := List( arg, Source );
return UniversalMorphismFromCoproductOp( diagram, arg, diagram[1] );
end );
####################################
## Categorical methods
####################################
##
InstallMethod( InjectionOfCofactorOfCoproduct,
[ IsList, IsInt ],
function( object_product_list, injection_number )
return InjectionOfCofactorOfCoproductOp( object_product_list, injection_number, object_product_list[1] );
end );
####################################
## Functorial operations
####################################
##
InstallMethod( CoproductFunctorial,
[ IsList ],
function( morphism_list )
return CoproductFunctorialWithGivenCoproducts(
Coproduct( List( morphism_list, Source ) ),
morphism_list,
Coproduct( List( morphism_list, Range ) )
);
end );
####################################
##
## Direct Product and FiberProduct
##
####################################
####################################
## Convenience methods
####################################
####################################
##
## Direct Product
##
####################################
####################################
## Convenience methods
####################################
##
InstallGlobalFunction( UniversalMorphismIntoDirectProduct,
function( arg )
local diagram;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismIntoDirectProductOp( arg[1], arg[2], arg[1][1] );
elif Length( arg ) = 1 and IsList( arg[ 1 ] ) then
arg := arg[ 1 ];
fi;
##convenience: UniversalMorphismIntoDirectProduct( test_projection_1, ..., test_projection_k )
diagram := List( arg, Range );
return UniversalMorphismIntoDirectProductOp( diagram, arg, diagram[1] );
end );
####################################
## Categorical methods
####################################
##
InstallMethod( ProjectionInFactorOfDirectProduct,
[ IsList, IsInt ],
function( object_product_list, projection_number )
return ProjectionInFactorOfDirectProductOp( object_product_list, projection_number, object_product_list[1] );
end );
####################################
## Functorial operations
####################################
##
InstallMethod( DirectProductFunctorial,
[ IsList ],
function( morphism_list )
return DirectProductFunctorialWithGivenDirectProducts(
DirectProduct( List( morphism_list, Source ) ),
morphism_list,
DirectProduct( List( morphism_list, Range ) ) );
end );
####################################
##
## Direct sum
##
####################################
####################################
## Technical methods
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( ProjectionInFactorOfDirectSum,
[ IsList, IsInt ],
function( diagram, projection_number )
return ProjectionInFactorOfDirectSumOp( diagram, projection_number, diagram[1] );
end );
##
InstallMethod( InjectionOfCofactorOfDirectSum,
[ IsList, IsInt ],
function( diagram, injection_number )
return InjectionOfCofactorOfDirectSumOp( diagram, injection_number, diagram[1] );
end );
##
InstallGlobalFunction( UniversalMorphismFromDirectSum,
function( arg )
local diagram;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismFromDirectSumOp( arg[1], arg[2], arg[1][1] );
elif Length( arg ) = 1 and IsList( arg[ 1 ] ) then
arg := arg[ 1 ];
fi;
diagram := List( arg, Source );
return UniversalMorphismFromDirectSumOp( diagram, arg, diagram[1] );
end );
##
InstallGlobalFunction( UniversalMorphismIntoDirectSum,
function( arg )
local diagram;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismIntoDirectSumOp( arg[1], arg[2], arg[1][1] );
elif Length( arg ) = 1 and IsList( arg[ 1 ] ) then
arg := arg[ 1 ];
fi;
##convenience: UniversalMorphismIntoDirectSum( test_projection_1, ..., test_projection_k )
diagram := List( arg, Range );
return UniversalMorphismIntoDirectSumOp( diagram, arg, diagram[1] );
end );
##
InstallMethod( IsomorphismFromDirectProductToDirectSum,
[ IsList ],
function( diagram )
return IsomorphismFromDirectProductToDirectSumOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromDirectSumToDirectProduct,
[ IsList ],
function( diagram )
return IsomorphismFromDirectSumToDirectProductOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromCoproductToDirectSum,
[ IsList ],
function( diagram )
return IsomorphismFromCoproductToDirectSumOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromDirectSumToCoproduct,
[ IsList ],
function( diagram )
return IsomorphismFromDirectSumToCoproductOp( diagram, diagram[1] );
end );
##
InstallMethod( DirectSumOp,
[ IsList, IsCapCategory ],
function( category_and_diagram, category )
local diagram;
diagram := category_and_diagram[2];
if diagram = [ ] then
return ZeroObject( category );
fi;
return DirectSumOp( diagram, diagram[1] );
end );
####################################
## Functorial operations
####################################
##
InstallMethod( DirectSumFunctorial,
[ IsList ],
function( diagram )
return DirectSumFunctorialWithGivenDirectSums(
DirectSum( List( diagram, Source ) ), diagram, DirectSum( List( diagram, Range ) )
);
end );
####################################
## Add methods
####################################
####################################
## Categorical methods
####################################
##
InstallMethod( MorphismBetweenDirectSums,
[ IsList ],
function( morphism_matrix )
local morphism_matrix_listlist, row, rows, cols;
morphism_matrix_listlist := [ ];
for row in morphism_matrix do
Append( morphism_matrix_listlist, row );
od;
rows := Length( morphism_matrix );
cols := Length( morphism_matrix[1] );
return MorphismBetweenDirectSumsOp( morphism_matrix_listlist, rows, cols, morphism_matrix[1][1] );
end );
##
InstallMethod( MorphismBetweenDirectSums,
[ IsCapCategoryObject, IsList, IsCapCategoryObject ],
function( S, morphism_matrix, T )
if morphism_matrix = [ ] or morphism_matrix[1] = [ ] then
return ZeroMorphism( S, T );
fi;
return MorphismBetweenDirectSums( morphism_matrix );
end );
##
InstallMethodWithCacheFromObject( MorphismBetweenDirectSumsOp,
[ IsList, IsInt, IsInt, IsCapCategoryMorphism ],
function( morphism_matrix_listlist, rows, cols, caching_object )
local morphism_matrix, i, diagram_direct_sum_source, diagram_direct_sum_range, test_diagram_product, test_diagram_coproduct, morphism_into_product;
morphism_matrix := [ ];
for i in [ 1 .. rows ] do
Add( morphism_matrix, morphism_matrix_listlist{[(i-1)*cols + 1 .. i*cols]} );
od;
diagram_direct_sum_source := List( morphism_matrix, row -> Source( row[1] ) );
diagram_direct_sum_range := List( morphism_matrix[1], entry -> Range( entry ) );
test_diagram_coproduct := [ ];
for test_diagram_product in morphism_matrix do
Add( test_diagram_coproduct, UniversalMorphismIntoDirectSum( diagram_direct_sum_range, test_diagram_product ) );
od;
return UniversalMorphismFromDirectSum( diagram_direct_sum_source, test_diagram_coproduct );
end: ArgumentNumber := 4 );
####################################
##
## Zero Object
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( ZeroObject,
[ IsCapCategoryCell ],
function( cell )
return ZeroObject( CapCategory( cell ) );
end );
##
InstallMethod( MorphismFromZeroObject,
[ IsCapCategoryObject ],
function( obj )
return UniversalMorphismFromZeroObject( obj );
end );
##
InstallMethod( MorphismIntoZeroObject,
[ IsCapCategoryObject ],
function( obj )
return UniversalMorphismIntoZeroObject( obj );
end );
####################################
##
## Terminal Object
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( TerminalObject,
[ IsCapCategoryCell ],
function( cell )
return TerminalObject( CapCategory( cell ) );
end );
####################################
##
## Initial Object
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( InitialObject,
[ IsCapCategoryCell ],
function( cell )
return InitialObject( CapCategory( cell ) );
end );
####################################
##
## FiberProduct
##
####################################
####################################
## Convenience methods
####################################
##
InstallGlobalFunction( FiberProduct,
function( arg )
if Length( arg ) = 1
and IsList( arg[1] )
and ForAll( arg[1], IsCapCategoryMorphism ) then
return FiberProductOp( arg[1], arg[1][1] );
fi;
return FiberProductOp( arg, arg[ 1 ] );
end );
##
InstallMethod( ProjectionInFactorOfFiberProduct,
[ IsList, IsInt ],
function( diagram, projection_number )
return ProjectionInFactorOfFiberProductOp( diagram, projection_number, diagram[1] );
end );
##
InstallGlobalFunction( UniversalMorphismIntoFiberProduct,
function( arg )
local diagram, source;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismIntoFiberProductOp( arg[1], arg[2], arg[1][1] );
fi;
diagram := arg[ 1 ];
source := arg{[ 2 .. Length( arg ) ]};
return UniversalMorphismIntoFiberProductOp( diagram, source, diagram[1] );
end );
##
InstallMethod( DirectSumDiagonalDifference,
[ IsList ],
function( diagram )
return DirectSumDiagonalDifferenceOp( diagram, diagram[1] );
end );
##
InstallMethod( FiberProductEmbeddingInDirectSum,
[ IsList ],
function( diagram )
return FiberProductEmbeddingInDirectSumOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromFiberProductToKernelOfDiagonalDifference,
[ IsList ],
function( diagram )
return IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct,
[ IsList ],
function( diagram )
return IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromPushoutToCokernelOfDiagonalDifference,
[ IsList ],
function( diagram )
return IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp( diagram, diagram[1] );
end );
##
InstallMethod( IsomorphismFromCokernelOfDiagonalDifferenceToPushout,
[ IsList ],
function( diagram )
return IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp( diagram, diagram[1] );
end );
####################################
## Functorial operations
####################################
##
InstallMethod( FiberProductFunctorial,
[ IsList ],
function( morphism_of_morphisms )
return FiberProductFunctorialWithGivenFiberProducts(
FiberProduct( List( morphism_of_morphisms, elem -> elem[1] ) ),
morphism_of_morphisms,
FiberProduct( List( morphism_of_morphisms, elem -> elem[3] ) )
);
end );
####################################
##
## Pushout
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( Pushout,
[ IsList ],
function( diagram )
return PushoutOp( diagram, diagram[1] );
end );
##
InstallMethod( Pushout,
[ IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( mor1, mor2 )
return PushoutOp( [ mor1, mor2 ], mor1 );
end );
##
InstallMethod( InjectionOfCofactorOfPushout,
[ IsList, IsInt ],
function( diagram, injection_number )
return InjectionOfCofactorOfPushoutOp( diagram, injection_number, diagram[1] );
end );
##
InstallGlobalFunction( UniversalMorphismFromPushout,
function( arg )
local diagram, sink;
if Length( arg ) = 2
and IsList( arg[1] )
and IsList( arg[2] ) then
return UniversalMorphismFromPushoutOp( arg[1], arg[2], arg[1][1] );
fi;
diagram := arg[ 1 ];
sink := arg{[ 2 .. Length( arg ) ]};
return UniversalMorphismFromPushoutOp( diagram, sink, diagram[1] );
end );
##
InstallMethod( DirectSumCodiagonalDifference,
[ IsList ],
function( diagram )
return DirectSumCodiagonalDifferenceOp( diagram, diagram[1] );
end );
##
InstallMethod( DirectSumProjectionInPushout,
[ IsList ],
function( diagram )
return DirectSumProjectionInPushoutOp( diagram, diagram[1] );
end );
####################################
## Functorial operations
####################################
##
InstallMethod( PushoutFunctorial,
[ IsList ],
function( morphism_of_morphisms )
return PushoutFunctorialWithGivenPushouts(
Pushout( List( morphism_of_morphisms, elem -> elem[1] ) ),
morphism_of_morphisms,
Pushout( List( morphism_of_morphisms, elem -> elem[3] ) )
);
end );
####################################
##
## Coimage
##
####################################
####################################
## Convenience methods
####################################
##
InstallMethod( MorphismFromCoimageToImage,
[ IsCapCategoryMorphism ],
function( morphism )
return MorphismFromCoimageToImageWithGivenObjects( Coimage( morphism ), morphism, ImageObject( morphism ) );
end );
##
InstallMethod( InverseMorphismFromCoimageToImage,
[ IsCapCategoryMorphism ],
function( morphism )
return InverseMorphismFromCoimageToImageWithGivenObjects( Coimage( morphism ), morphism, ImageObject( morphism ) );
end );
####################################
##
## Scheme for Universal Object
##
####################################
####################################
## Add Operations
####################################
####################################
## Attributes
####################################
####################################
## Implied Operations
####################################