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# SPDX-License-Identifier: GPL-2.0-or-later
# GeneralizedMorphismsForCAP: Implementations of generalized morphisms for the CAP project
#
# Implementations
#
#############################################
##
## Installation method
##
#############################################
BindGlobal( "CAP_INTERNAL_INSTALL_OPERATIONS_FOR_SERRE_QUOTIENT_BY_COSPANS",
function( category )
local membership_function;
membership_function := SubcategoryMembershipTestFunctionForSerreQuotient( category );
## Equalities
AddIsCongruentForMorphisms( category,
function( cat, morphism1, morphism2 )
local underlying_generalized_category, underlying_general, new_morphism_aid, new_general, sum_general,
sum_associated, sum_image;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
underlying_general := UnderlyingGeneralizedMorphism( morphism2 );
new_general := AdditiveInverse( underlying_generalized_category, underlying_general );
sum_general := AdditionForMorphisms( underlying_generalized_category, UnderlyingGeneralizedMorphism( morphism1 ), new_general );
sum_associated := AssociatedMorphism( sum_general );
sum_image := ImageObject( underlying_generalized_category, sum_associated );
return membership_function( sum_image );
end );
AddIsEqualForObjects( category,
function( cat, obj1, obj2 )
return IsEqualForObjects( UnderlyingHonestCategory( cat ), UnderlyingHonestObject( obj1 ), UnderlyingHonestObject( obj2 ) );
end );
## Is Zero
AddIsZeroForObjects( category,
function( cat, obj )
return membership_function( UnderlyingHonestObject( obj ) );
end );
## PreCompose
AddPreCompose( category,
function( cat, morphism1, morphism2 )
local underlying_generalized_category, composition;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
composition := PreCompose( underlying_generalized_category,
UnderlyingGeneralizedMorphism( morphism1 ),
UnderlyingGeneralizedMorphism( morphism2 ) );
return SerreQuotientCategoryByCospansMorphism( cat, composition );
end );
## IdentityMorphism
AddIdentityMorphism( category,
function( cat, object )
return AsSerreQuotientCategoryByCospansMorphism( cat,
IdentityMorphism( UnderlyingHonestCategory( cat ),
UnderlyingHonestObject( object ) ) );
end );
## Addition for morphisms
AddAdditionForMorphisms( category,
function( cat, morphism1, morphism2 )
local underlying_generalized_category, sum;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
sum := AdditionForMorphisms( underlying_generalized_category,
UnderlyingGeneralizedMorphism( morphism1 ),
UnderlyingGeneralizedMorphism( morphism2 ) );
return SerreQuotientCategoryByCospansMorphism( cat, sum );
end );
## IsZeroForMorphisms
## Can be derived, but there might be a faster solution
# AddIsZeroForMorphisms( category,
#
# function( cat, morphism )
# local honest_category, associated, image;
#
# honest_category := UnderlyingHonestCategory( cat );
#
# associated := AssociatedMorphism( UnderlyingGeneralizedMorphism( morphism ) );
#
# image := ImageObject( honest_category, associated );
#
# return membership_function( image );
#
# end );
## Additive inverse for morphisms (works without normalization)
AddAdditiveInverseForMorphisms( category,
function( cat, morphism )
local underlying_generalized_category, new_general;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
new_general := AdditiveInverseForMorphisms( underlying_generalized_category, UnderlyingGeneralizedMorphism( morphism ) );
return SerreQuotientCategoryByCospansMorphism( cat, new_general );
end );
## Zero morphism
AddZeroMorphism( category,
function( cat, source, range )
local underlying_generalized_category, new_general;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
new_general := ZeroMorphism( underlying_generalized_category,
UnderlyingGeneralizedObject( source ),
UnderlyingGeneralizedObject( range ) );
return SerreQuotientCategoryByCospansMorphism( cat, new_general );
end );
## Zero object
AddZeroObject( category,
function( cat )
local generalized_zero;
generalized_zero := ZeroObject( UnderlyingHonestCategory( cat ) );
return AsSerreQuotientCategoryByCospansObject( cat, generalized_zero );
end );
## direct sum
AddDirectSum( category,
function( cat, obj_list )
local honest_category, honest_list, honest_sum;
honest_category := UnderlyingHonestCategory( cat );
honest_list := List( obj_list, UnderlyingHonestObject );
honest_sum := DirectSum( cat, honest_list );
return AsSerreQuotientCategoryByCospansObject( cat, honest_sum );
end );
AddProjectionInFactorOfDirectSumWithGivenDirectSum( category,
function( cat, product_object, component_number, direct_sum_object )
local honest_category, underlying_objects, underlying_direct_sum, honest_projection;
honest_category := UnderlyingHonestCategory( cat );
underlying_objects := List( product_object, UnderlyingHonestObject );
underlying_direct_sum := UnderlyingHonestObject( direct_sum_object );
honest_projection := ProjectionInFactorOfDirectSumWithGivenDirectSum( honest_category,
underlying_objects,
component_number,
underlying_direct_sum );
return AsSerreQuotientCategoryByCospansMorphism( cat, honest_projection );
end );
AddInjectionOfCofactorOfDirectSumWithGivenDirectSum( category,
function( cat, object_product_list, injection_number, direct_sum_object )
local honest_category, underlying_objects, underlying_direct_sum, honest_injection;
honest_category := UnderlyingHonestCategory( cat );
underlying_objects := List( object_product_list, UnderlyingHonestObject );
underlying_direct_sum := UnderlyingHonestObject( direct_sum_object );
honest_injection := AddInjectionOfCofactorOfDirectSumWithGivenDirectSum( honest_category,
underlying_objects,
injection_number,
underlying_direct_sum );
return AsSerreQuotientCategoryByCospansMorphism( cat, honest_injection );
end );
AddUniversalMorphismIntoDirectSum( category,
function( cat, diagram, test_object, morphism_list )
local honest_category, generalized_list, arrow_list, reversedarrow_list, new_arrow, new_reversed_arrow, object_list;
honest_category := UnderlyingHonestCategory( cat );
generalized_list := List( morphism_list, UnderlyingGeneralizedMorphism );
arrow_list := List( generalized_list, Arrow );
new_arrow := UniversalMorphismIntoDirectSum( honest_category, List( arrow_list, Range ), arrow_list );
reversedarrow_list := List( generalized_list, ReversedArrow );
new_reversed_arrow := DirectSumFunctorial( honest_category, reversedarrow_list );
return SerreQuotientCategoryByCospansMorphism( cat, new_arrow, new_reversed_arrow );
end );
## Needs work
AddUniversalMorphismFromDirectSum( category,
function( cat, diagram, test_object, morphism_list )
local honest_category, generalized_list, arrow_list, reversedarrow_list, new_arrow, new_reversed_arrow, object_list;
honest_category := UnderlyingHonestCategory( cat );
generalized_list := List( morphism_list, UnderlyingGeneralizedMorphism );
generalized_list := CommonCoastriction( generalized_list );
arrow_list := List( generalized_list, Arrow );
new_arrow := UniversalMorphismFromDirectSum( honest_category, List( diagram, UnderlyingHonestObject ), arrow_list );
new_reversed_arrow := ReversedArrow( generalized_list[ 1 ] );
return SerreQuotientCategoryByCospansMorphism( cat, new_arrow, new_reversed_arrow );
end );
## Kernel
AddKernelEmbedding( category,
function( cat, morphism )
local honest_category, underlying_general, kernel_mor;
honest_category := UnderlyingHonestCategory( cat );
underlying_general := UnderlyingGeneralizedMorphism( morphism );
kernel_mor := KernelEmbedding( honest_category, Arrow( underlying_general ) );
return AsSerreQuotientCategoryByCospansMorphism( cat, kernel_mor );
end );
AddLiftAlongMonomorphism( category,
function( cat, monomorphism, test_morphism )
local underlying_generalized_category, inverse_of_mono, composition;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
inverse_of_mono := PseudoInverse( UnderlyingGeneralizedMorphism( monomorphism ) );
composition := PreCompose( underlying_generalized_category, UnderlyingGeneralizedMorphism( test_morphism ), inverse_of_mono );
return SerreQuotientCategoryByCospansMorphism( cat, composition );
end );
## Cokernel
AddCokernelProjection( category,
function( cat, morphism )
local honest_category, underlying_general, cokernel_mor, triple;
honest_category := UnderlyingHonestCategory( cat );
underlying_general := UnderlyingGeneralizedMorphism( morphism );
triple := DomainAssociatedMorphismCodomainTriple( underlying_general );
cokernel_mor := CokernelProjection( honest_category, triple[ 2 ] );
return AsSerreQuotientCategoryByCospansMorphism( cat, PreCompose( honest_category, triple[ 3 ], cokernel_mor ) );
end );
AddColiftAlongEpimorphism( category,
function( cat, epimorphism, test_morphism )
local underlying_generalized_category, inverse_of_epi, composition;
underlying_generalized_category := UnderlyingGeneralizedMorphismCategory( cat );
inverse_of_epi := PseudoInverse( UnderlyingGeneralizedMorphism( epimorphism ) );
composition := PreCompose( underlying_generalized_category, inverse_of_epi, UnderlyingGeneralizedMorphism( test_morphism ) );
return SerreQuotientCategoryByCospansMorphism( cat, composition );
end );
end );
#############################################
##
## Constructor
##
#############################################
InstallMethod( SerreQuotientCategoryByCospans,
[ IsCapCategory, IsFunction ],
function( category, test_function )
local name;
name := NameFunction( test_function );
return SerreQuotientCategoryByCospans( category, test_function, Concatenation( "test function with name: ", name ) );
end );
InstallMethodWithCacheFromObject( SerreQuotientCategoryByCospans,
[ IsCapCategory, IsFunction, IsString ],
function( category, test_function, function_name )
local serre_category, gen_category, name, preconditions, i;
if not IsFinalized( category ) then
Error( "category must be finalized" );
return;
fi;
preconditions := [ "IsEqualForObjects",
"ImageObject",
"AdditiveInverseForMorphisms",
"IdentityMorphism",
"ZeroMorphism",
"DirectSum",
"ProjectionInFactorOfDirectSumWithGivenDirectSum",
"InjectionOfCofactorOfDirectSumWithGivenDirectSum",
"UniversalMorphismFromDirectSum",
"UniversalMorphismIntoDirectSum",
"DirectSumFunctorialWithGivenDirectSums",
"KernelEmbedding",
"CokernelProjection" ];
for i in preconditions do
if not CanCompute( category, i ) then
Error( Concatenation( "category must be able to compute ", i ) );
return;
fi;
od;
name := Name( category );
name := Concatenation( "The Serre quotient category of ", name, " by ", function_name );
serre_category := CreateCapCategoryWithDataTypes(
name,
IsSerreQuotientCategory,
IsSerreQuotientCategoryByCospansObject,
IsSerreQuotientCategoryByCospansMorphism and HasUnderlyingGeneralizedMorphism,
IsCapCategoryTwoCell,
fail,
fail,
fail );
serre_category!.predicate_logic := category!.predicate_logic;
SetFilterObj( serre_category, WasCreatedAsSerreQuotientCategoryByCospans );
SetUnderlyingHonestCategory( serre_category, category );
SetUnderlyingGeneralizedMorphismCategory( serre_category, GeneralizedMorphismCategoryByCospans( category ) );
SetSubcategoryMembershipTestFunctionForSerreQuotient( serre_category, test_function );
SetIsAbelianCategory( serre_category, true );
CAP_INTERNAL_INSTALL_OPERATIONS_FOR_SERRE_QUOTIENT_BY_COSPANS( serre_category );
CAP_INTERNAL_INSTALL_OPERATIONS_FOR_SERRE_QUOTIENT( serre_category );
Finalize( serre_category );
return serre_category;
end );
InstallMethodWithCacheFromObject( AsSerreQuotientCategoryByCospansObject,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryObject ],
function( serre_category, object )
local honest_category, serre_object;
honest_category := UnderlyingHonestCategory( serre_category );
if not IsIdenticalObj( honest_category, CapCategory( object ) ) then
Error( "object does not belong to underlying honest category of serre quotient" );
fi;
serre_object := CreateCapCategoryObjectWithAttributes( serre_category,
UnderlyingHonestObject, object,
UnderlyingGeneralizedObject, GeneralizedMorphismByCospansObject( object ) );
if HasSpecializedObjectFilterForSerreQuotients( serre_category ) then
SetFilterObj( serre_object, SpecializedObjectFilterForSerreQuotients( serre_category ) );
fi;
return serre_object;
end );
InstallMethodWithCacheFromObject( SerreQuotientCategoryByCospansMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsGeneralizedMorphismByCospan ],
function( serre_category, gen_morphism )
local honest_category, serre_morphism;
if not IsIdenticalObj( UnderlyingGeneralizedMorphismCategory( serre_category ), CapCategory( gen_morphism ) ) then
Error( "generalized morphism has wrong category" );
fi;
serre_morphism := CreateCapCategoryMorphismWithAttributes( serre_category,
AsSerreQuotientCategoryByCospansObject( serre_category, UnderlyingHonestObject( Source( gen_morphism ) ) ),
AsSerreQuotientCategoryByCospansObject( serre_category, UnderlyingHonestObject( Range( gen_morphism ) ) ),
UnderlyingGeneralizedMorphism, gen_morphism );
if HasSpecializedMorphismFilterForSerreQuotients( serre_category ) then
SetFilterObj( serre_morphism, SpecializedMorphismFilterForSerreQuotients( serre_category ) );
fi;
return serre_morphism;
end );
InstallMethod( SerreQuotientCategoryByCospansMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( serre_category, source_aid, associated, range_aid )
return SerreQuotientCategoryByCospansMorphism( serre_category, GeneralizedMorphismByCospan( source_aid, associated, range_aid ) );
end );
InstallMethod( SerreQuotientCategoryByCospansMorphismWithSourceAid,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( serre_category, source_aid, associated )
return SerreQuotientCategoryByCospansMorphism( serre_category, GeneralizedMorphismByCospanWithSourceAid( source_aid, associated ) );
end );
InstallMethod( SerreQuotientCategoryByCospansMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( serre_category, reversed_arrow, arrow )
return SerreQuotientCategoryByCospansMorphism( serre_category, GeneralizedMorphismByCospan( reversed_arrow, arrow ) );
end );
InstallMethodWithCacheFromObject( AsSerreQuotientCategoryByCospansMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism ],
function( serre_category, associated )
return SerreQuotientCategoryByCospansMorphism( serre_category, AsGeneralizedMorphismByCospan( associated ) );
end );
#############################################
##
## Compatibility layer
##
#############################################
InstallMethod( AsSerreQuotientCategoryObject,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryObject ],
AsSerreQuotientCategoryByCospansObject );
InstallMethod( AsSerreQuotientCategoryMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism ],
AsSerreQuotientCategoryByCospansMorphism );
InstallMethod( SerreQuotientCategoryMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsGeneralizedMorphismByCospan ],
SerreQuotientCategoryByCospansMorphism );
InstallMethod( SerreQuotientCategoryMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ],
SerreQuotientCategoryByCospansMorphism );
InstallMethod( SerreQuotientCategoryMorphism,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism ],
SerreQuotientCategoryByCospansMorphism );
InstallMethod( SerreQuotientCategoryMorphismWithRangeAid,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism ],
SerreQuotientCategoryByCospansMorphism );
InstallMethod( SerreQuotientCategoryMorphismWithSourceAid,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans, IsCapCategoryMorphism, IsCapCategoryMorphism ],
SerreQuotientCategoryByCospansMorphismWithSourceAid );
#############################################
##
## Functor
##
#############################################
InstallMethod( CanonicalProjection,
[ IsCapCategory and WasCreatedAsSerreQuotientCategoryByCospans ],
function( category )
local underlying_honest, functor;
underlying_honest := UnderlyingHonestCategory( category );
functor := CapFunctor( Concatenation( "Localization functor of ", Name( category ) ), underlying_honest, category );
AddObjectFunction( functor,
i -> AsSerreQuotientCategoryByCospansObject( category, i ) );
AddMorphismFunction( functor,
function( new_source, morphism, new_range )
return AsSerreQuotientCategoryByCospansMorphism( category, morphism );
end );
return functor;
end );
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