LIMOR.gi
# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## LIMOR = Logical Implications for homalg MORphisms
####################################
#
# global variables:
#
####################################
# a central place for configuration variables:
InstallValue( LIMOR,
rec(
color := "\033[4;30;46m",
intrinsic_properties :=
[ "IsZero",
"IsMorphism",
"IsGeneralizedMorphismWithFullDomain",
"IsGeneralizedEpimorphism",
"IsGeneralizedMonomorphism",
"IsGeneralizedIsomorphism",
"IsOne",
"IsIdempotent",
"IsMonomorphism",
"IsEpimorphism",
"IsSplitMonomorphism",
"IsSplitEpimorphism",
"IsIsomorphism",
"IsAutomorphism" ],
intrinsic_attributes :=
[ "DegreeOfMorphism" ],
)
);
##
InstallValue( LogicalImplicationsForHomalgMorphisms,
[
## IsZero does not imply IsMorphism!!!
[ IsZero,
"implies", IsGeneralizedMorphismWithFullDomain ],
[ IsMorphism,
"implies", IsGeneralizedMorphismWithFullDomain ],
# Comply with CAP: no mathematical properties should imply IsMorphism
# [ IsMonomorphism,
# "implies", IsMorphism ],
[ IsMonomorphism,
"implies", IsGeneralizedMonomorphism ],
[ IsGeneralizedMonomorphism,
"implies", IsGeneralizedMorphismWithFullDomain ],
# Comply with CAP: no mathematical properties should imply IsMorphism
# [ IsEpimorphism,
# "implies", IsMorphism ],
[ IsEpimorphism,
"implies", IsGeneralizedEpimorphism ],
[ IsGeneralizedEpimorphism,
"implies", IsGeneralizedMorphismWithFullDomain ],
[ IsAutomorphism,
"implies", IsIsomorphism ],
[ IsIsomorphism,
"implies", IsGeneralizedIsomorphism ],
[ IsIsomorphism,
"implies", IsSplitMonomorphism ],
[ IsIsomorphism,
"implies", IsSplitEpimorphism ],
[ IsSplitEpimorphism,
"implies", IsEpimorphism ],
[ IsSplitMonomorphism,
"implies", IsMonomorphism ],
[ IsEpimorphism, "and", IsMonomorphism,
"imply", IsIsomorphism ],
[ IsGeneralizedIsomorphism,
"implies", IsGeneralizedMonomorphism ],
[ IsGeneralizedIsomorphism,
"implies", IsGeneralizedEpimorphism ],
[ IsGeneralizedEpimorphism, "and", IsGeneralizedMonomorphism,
"imply", IsGeneralizedIsomorphism ],
## this is wrong:
## [ IsGeneralizedEpimorphism, "and", IsMorphism,
## "imply", IsEpimorphism ],
[ IsGeneralizedMonomorphism, "and", IsMorphism,
"imply", IsMonomorphism ],
[ IsOne,
"implies", IsAutomorphism ],
] );
##
InstallValue( LogicalImplicationsForHomalgEndomorphisms,
[
[ IsZero, "and", IsMorphism,
"imply", IsIdempotent ],
[ IsOne, "and", IsMorphism,
"imply", IsIdempotent ],
[ IsIsomorphism,
"implies", IsAutomorphism ],
] );
####################################
#
# logical implications methods:
#
####################################
InstallLogicalImplicationsForHomalgObjects( LogicalImplicationsForHomalgMorphisms, IsHomalgMorphism );
InstallLogicalImplicationsForHomalgObjects( LogicalImplicationsForHomalgEndomorphisms, IsHomalgEndomorphism );
####################################
#
# immediate methods for properties:
#
####################################
##
InstallImmediateMethod( IsZero,
IsHomalgMorphism, 0,
function( phi )
if ( HasIsZero( Source( phi ) ) and IsZero( Source( phi ) ) ) and ## to place "or" here we need to know that phi is a morphism;
( HasIsZero( Range( phi ) ) and IsZero( Range( phi ) ) ) then ## see the method below
return true;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsZero,
IsHomalgMorphism and IsMorphism, 0,
function( phi )
if ( HasIsZero( Source( phi ) ) and IsZero( Source( phi ) ) ) or
( HasIsZero( Range( phi ) ) and IsZero( Range( phi ) ) ) then
return true;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsZero,
IsHomalgMorphism and IsMonomorphism, 0,
function( phi )
local S;
S := Source( phi );
if HasIsZero( S ) then
return IsZero( S );
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsZero,
IsHomalgMorphism and IsEpimorphism, 0,
function( phi )
local T;
T := Range( phi );
if HasIsZero( T ) then
return IsZero( T );
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsMorphism,
IsHomalgMorphism and IsZero, 0,
function( phi )
if HasMorphismAid( phi ) then
TryNextMethod( );
fi;
return true;
end );
##
InstallImmediateMethod( IsSplitEpimorphism,
IsHomalgStaticMorphism and IsEpimorphism, 0,
function( phi )
local T;
T := Range( phi );
if HasIsProjective( T ) and IsProjective( T ) then
return true;
fi;
TryNextMethod( );
end );
##
## this immediate method together with the above ture-method
## IsZero => IsMorphism
## are essential to avoid infinite loops that
## Assert( 5, IsMonomorphism( emb ) );
## in _Functor_ImageObject_OnObjects may cause
##
InstallImmediateMethod( IsSplitMonomorphism,
IsHomalgMorphism and IsMorphism, 0,
function( phi )
local S;
S := Source( phi );
if HasIsZero( S ) and IsZero( S ) then
return true;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsAutomorphism,
IsHomalgMorphism, 0,
function( phi )
if not IsIdenticalObj( Source( phi ), Range( phi ) ) then
return false;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsIdempotent,
IsHomalgMorphism, 0,
function( phi )
if not IsIdenticalObj( Source( phi ), Range( phi ) ) then
return false;
fi;
TryNextMethod( );
end );
####################################
#
# methods for properties:
#
####################################
##
InstallMethod( IsMorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and HasMorphismAid ], 10001,
function( phi )
if not IsZero( MorphismAid( phi ) ) then
return false;
fi;
TryNextMethod( );
end );
##
InstallMethod( IsGeneralizedMorphismWithFullDomain,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
if HasMorphismAid( phi ) then
TryNextMethod( );
fi;
return IsMorphism( phi );
end );
##
InstallMethod( IsGeneralizedMorphismWithFullDomain,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and HasMorphismAid ],
function( phi )
return IsMorphism( AssociatedMorphismOfGeneralizedMorphismWithFullDomain( phi ) );
end );
##
InstallMethod( IsEpimorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
local b, S, T;
b := IsMorphism( phi ) and IsZero( Cokernel( phi ) );
if b then
S := Source( phi );
T := Range( phi );
if HasIsTorsion( T ) and not IsTorsion( T ) then
SetIsTorsion( S, false );
elif HasIsTorsion( S ) and IsTorsion( S ) then
SetIsTorsion( T, true );
fi;
fi;
return b;
end );
##
InstallMethod( IsGeneralizedEpimorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
## this is just the fall back method
return IsEpimorphism( phi );
end );
##
InstallMethod( IsGeneralizedEpimorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and HasMorphismAid ],
function( phi )
local mu;
mu := AssociatedMorphismOfGeneralizedMorphismWithFullDomain( phi );
SetIsGeneralizedMorphismWithFullDomain( phi, IsMorphism( mu ) );
return IsEpimorphism( mu );
end );
##
InstallMethod( IsSplitEpimorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
local split;
if not IsEpimorphism( phi ) then
return false;
fi;
split := PreInverse( phi );
if split = fail then
TryNextMethod( );
fi;
return split <> false;
end );
##
InstallMethod( IsMonomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
local b, S, T;
b := IsMorphism( phi ) and IsZero( Kernel( phi ) );
if b then
S := Source( phi );
T := Range( phi );
if HasIsTorsionFree( T ) and IsTorsionFree( T ) then
SetIsTorsionFree( S, true );
elif HasIsTorsionFree( S ) and not IsTorsionFree( S ) then
SetIsTorsionFree( T, false );
fi;
if HasIsTorsion( T ) and IsTorsion( T ) then
SetIsTorsion( S, true );
elif HasIsTorsion( S ) and not IsTorsion( S ) then
SetIsTorsion( T, false );
fi;
fi;
return b;
end );
##
InstallMethod( IsGeneralizedMonomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
## this is just the fall back method
return IsMonomorphism( phi );
end );
##
InstallMethod( IsGeneralizedMonomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and HasMorphismAid ],
function( phi )
local mu;
mu := AssociatedMorphismOfGeneralizedMorphismWithFullDomain( phi );
SetIsGeneralizedMorphismWithFullDomain( phi, IsMorphism( mu ) );
return IsMonomorphism( mu );
end );
##
InstallMethod( IsIsomorphism,
"LIMOR: for homalg morphisms",
[ IsHomalgMorphism ],
function( phi )
local iso;
iso := IsEpimorphism( phi ) and IsMonomorphism( phi );
if iso then
SetIsIsomorphism( phi, true ); ## needed for UpdateObjectsByMorphism
UpdateObjectsByMorphism( phi );
fi;
return iso;
end );
##
InstallMethod( IsGeneralizedIsomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
## this is just the fall back method
return IsIsomorphism( phi );
end );
##
InstallMethod( IsGeneralizedIsomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and HasMorphismAid ],
function( phi )
local mu;
mu := AssociatedMorphismOfGeneralizedMorphismWithFullDomain( phi );
SetIsGeneralizedMorphismWithFullDomain( phi, IsMorphism( mu ) );
SetIsGeneralizedMonomorphism( phi, IsMonomorphism( mu ) );
return IsIsomorphism( mu );
end );
##
InstallMethod( IsAutomorphism,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism ],
function( phi )
return IsHomalgEndomorphism( phi ) and IsIsomorphism( phi );
end );
##
InstallMethod( IsIdempotent,
"LIMOR: for homalg morphisms",
[ IsHomalgMorphism ],
function( phi )
return IsHomalgEndomorphism( phi ) and IsZero( phi^2 - phi );
end );
####################################
#
# methods for attributes:
#
####################################
## Cf. [Bar, Cor. 4.8]
InstallMethod( InverseOfGeneralizedMorphismWithFullDomain,
"LIMOR: for morphisms of homalg static objects",
[ IsHomalgStaticMorphism and IsEpimorphism ],
function( epsilon )
local gen_iso, aid;
## the generalized inverse of the epimorphism
gen_iso := epsilon^-1;
## the morphism aid map of the generalized inverse
aid := MorphismHavingSubobjectAsItsImage( KernelSubobject( epsilon ) );
## set the morphism aid map
gen_iso := AddToMorphismAid( gen_iso, aid );
## check assertion
Assert( 5, IsGeneralizedIsomorphism( gen_iso ) );
SetIsGeneralizedIsomorphism( gen_iso, true );
return gen_iso;
end );
##
InstallMethod( Cokernel,
"for homalg static morphisms",
[ IsStaticMorphismOfFinitelyGeneratedObjectsRep and HasDirectSummands ], 1001,
function( phi )
local phis, coker, epi;
phis := DirectSummands( phi );
coker := List( phis, Cokernel );
coker := DirectSum( coker );
epi := List( phis, CokernelEpi );
epi := DiagonalMorphism( epi );
SetCokernelEpi( phi, epi );
return coker;
end );
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( SetPropertiesOfGeneralizedMorphism,
"for two homalg morphisms",
[ IsHomalgMorphism,
IsHomalgMorphism ],
function( psi, phi )
homalgResetFilters( psi );
if HasIsZero( phi ) and IsZero( phi ) then
SetIsZero( psi, true );
fi;
## this is wrong:
## if HasIsIsomorphism( phi ) and IsIsomorphism( phi ) then
## SetIsGeneralizedIsomorphism( psi, true );
## fi;
if HasIsGeneralizedEpimorphism( phi ) and IsGeneralizedEpimorphism( phi ) then
SetIsGeneralizedEpimorphism( psi, true );
elif HasIsGeneralizedMorphismWithFullDomain( phi ) and IsGeneralizedMorphismWithFullDomain( phi ) then
SetIsGeneralizedMorphismWithFullDomain( psi, true );
fi;
return psi;
end );
##
InstallMethod( SetPropertiesOfAdditiveInverse,
"for two homalg morphisms",
[ IsHomalgMorphism,
IsHomalgMorphism ],
function( psi, phi )
if HasIsMorphism( phi ) then
SetIsMorphism( psi, IsMorphism( phi ) );
fi;
if HasMorphismAid( phi ) then
psi := AddToMorphismAid( psi, MorphismAid( phi ) );
fi;
if HasIsGeneralizedMorphismWithFullDomain( phi ) then
SetIsGeneralizedMorphismWithFullDomain( psi, IsGeneralizedMorphismWithFullDomain( phi ) );
fi;
if HasIsEpimorphism( phi ) then
SetIsEpimorphism( psi, IsEpimorphism( phi ) );
fi;
if HasIsMonomorphism( phi ) then
SetIsMonomorphism( psi, IsMonomorphism( phi ) );
fi;
return psi;
end );
