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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
Path: gap4r8 / pkg / ModulePresentationsForCAP-2017.09.09 / gap / ModulePresentationNaturalTransformations.gi
Views: 418346#############################################################################
##
## ModulePresentationsForCAP package
##
## Copyright 2014, Sebastian Gutsche, TU Kaiserslautern
## Sebastian Posur, RWTH Aachen
##
#############################################################################
#################################
##
## Identity to StandardModule
##
#################################
BindGlobal( "INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_STANDARD_MODULE_METHOD",
function( natural_transformation_from_identity_to_standard_module_method, presentations, standard_module, nr_generators )
InstallMethod( natural_transformation_from_identity_to_standard_module_method,
[ IsHomalgRing ],
function( ring )
local standard_module_functor, category, natural_transformation;
category := presentations( ring );
standard_module_functor := standard_module( ring );
natural_transformation := NaturalTransformation( Concatenation( "Natural isomorphism from Id to ", Name( standard_module_functor ) ),
IdentityMorphism( AsCatObject( category ) ), standard_module_functor );
AddNaturalTransformationFunction( natural_transformation,
function( id_object, object, standard_object )
local matrix, natiso;
matrix := HomalgIdentityMatrix( nr_generators( UnderlyingMatrix( id_object ) ), ring );
natiso := PresentationMorphism( id_object, matrix, standard_object );
Assert( 4, IsIsomorphism( natiso ) );
SetIsIsomorphism( natiso, true );
return natiso;
end );
SetIsIsomorphism( natural_transformation, true );
return natural_transformation;
end );
end );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_STANDARD_MODULE_METHOD( NaturalIsomorphismFromIdentityToStandardModuleLeft,
LeftPresentations,
FunctorStandardModuleLeft,
NrColumns );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_STANDARD_MODULE_METHOD( NaturalIsomorphismFromIdentityToStandardModuleRight,
RightPresentations,
FunctorStandardModuleRight,
NrRows );
BindGlobal( "INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_GET_RID_OF_GENERATORS_METHOD",
function( natural_transformation_from_identity_to_get_rid_of_zero_generators_method, presentations, get_rid_of_zero_generators, triple_getter )
InstallMethod( natural_transformation_from_identity_to_get_rid_of_zero_generators_method,
[ IsHomalgRing ],
function( ring )
local get_rid_of_zero_generators_functor, category, natural_transformation;
category := presentations( ring );
get_rid_of_zero_generators_functor := get_rid_of_zero_generators( ring );
natural_transformation := NaturalTransformation( Concatenation( "Natural isomorphism from Id to ", Name( get_rid_of_zero_generators_functor ) ),
IdentityMorphism( AsCatObject( category ) ), get_rid_of_zero_generators_functor );
AddNaturalTransformationFunction( natural_transformation,
function( id_object, object, smaller_object )
local ZG, natiso;
ZG := triple_getter( UnderlyingMatrix( object ) );
natiso := PresentationMorphism( id_object, ZG[2], smaller_object );
Assert( 4, IsIsomorphism( natiso ) );
SetIsIsomorphism( natiso, true );
return natiso;
end );
SetIsIsomorphism( natural_transformation, true );
return natural_transformation;
end );
end );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_GET_RID_OF_GENERATORS_METHOD( NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft,
LeftPresentations,
FunctorGetRidOfZeroGeneratorsLeft,
NonZeroGeneratorsTransformationTripleLeft );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_GET_RID_OF_GENERATORS_METHOD( NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight,
RightPresentations,
FunctorGetRidOfZeroGeneratorsRight,
NonZeroGeneratorsTransformationTripleRight );
BindGlobal( "INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_LESS_GENERATORS_METHOD",
function( natural_transformation_from_identity_to_less_generators_method, presentations, less_generators, triple_getter )
InstallMethod( natural_transformation_from_identity_to_less_generators_method,
[ IsHomalgRing ],
function( ring )
local less_generators_functor, category, natural_transformation;
category := presentations( ring );
less_generators_functor := less_generators( ring );
natural_transformation := NaturalTransformation( Concatenation( "Natural isomorphism from Id to ", Name( less_generators_functor ) ),
IdentityMorphism( AsCatObject( category ) ), less_generators_functor );
AddNaturalTransformationFunction( natural_transformation,
function( id_object, object, smaller_object )
local LG, natiso;
LG := triple_getter( UnderlyingMatrix( object ) );
natiso := PresentationMorphism( id_object, LG[2], smaller_object );
Assert( 4, IsIsomorphism( natiso ) );
SetIsIsomorphism( natiso, true );
return natiso;
end );
SetIsIsomorphism( natural_transformation, true );
return natural_transformation;
end );
end );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_LESS_GENERATORS_METHOD( NaturalIsomorphismFromIdentityToLessGeneratorsLeft,
LeftPresentations,
FunctorLessGeneratorsLeft,
LessGeneratorsTransformationTripleLeft );
INSTALL_NATURAL_TRANSFORMATION_FROM_IDENTITY_TO_LESS_GENERATORS_METHOD( NaturalIsomorphismFromIdentityToLessGeneratorsRight,
RightPresentations,
FunctorLessGeneratorsRight,
LessGeneratorsTransformationTripleRight );
########################################################
##
## Identity to double dual left Hom_R( _ , R ) functor
##
########################################################
##
InstallMethod( NaturalTransformationFromIdentityToDoubleDualLeft,
[ IsHomalgRing ],
function( ring )
local category, double_dual_functor, natural_tasformation;
category := LeftPresentations( ring );
double_dual_functor := FunctorDoubleDualLeft( ring );
natural_tasformation := NaturalTransformation( Concatenation( "Natural transformation from Id to ", Name( double_dual_functor ) ),
IdentityMorphism( AsCatObject( category ) ), double_dual_functor );
AddNaturalTransformationFunction( natural_tasformation,
function( id_object, object, double_dual_object )
local A, representing_morphism, cokernel_projection_in_obj, double_dual_of_cokernel_projection, nat_mor;
A := UnderlyingMatrix( object );
representing_morphism := PresentationMorphism( FreeLeftPresentation( NrRows( A ), ring ), A, FreeLeftPresentation( NrColumns( A ), ring ) );
cokernel_projection_in_obj := CokernelProjection( representing_morphism );
double_dual_of_cokernel_projection := ApplyFunctor( double_dual_functor, cokernel_projection_in_obj );
nat_mor := CokernelColift( representing_morphism, double_dual_of_cokernel_projection );
return nat_mor;
end );
return natural_tasformation;
end );
##
InstallMethod( NaturalTransformationFromIdentityToDoubleDualRight,
[ IsHomalgRing ],
function( ring )
local category, double_dual_functor, natural_tasformation;
category := RightPresentations( ring );
double_dual_functor := FunctorDoubleDualRight( ring );
natural_tasformation := NaturalTransformation( Concatenation( "Natural transformation from Id to ", Name( double_dual_functor ) ),
IdentityMorphism( AsCatObject( category ) ), double_dual_functor );
AddNaturalTransformationFunction( natural_tasformation,
function( id_object, object, double_dual_object )
local A, representing_morphism, cokernel_projection_in_obj, double_dual_of_cokernel_projection, nat_mor;
A := UnderlyingMatrix( object );
representing_morphism := PresentationMorphism( FreeRightPresentation( NrColumns( A ), ring ), A, FreeRightPresentation( NrRows( A ), ring ) );
cokernel_projection_in_obj := CokernelProjection( representing_morphism );
double_dual_of_cokernel_projection := ApplyFunctor( double_dual_functor, cokernel_projection_in_obj );
nat_mor := CokernelColift( representing_morphism, double_dual_of_cokernel_projection );
return nat_mor;
end );
return natural_tasformation;
end );