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# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## Implementations for homalg elements.
## <#GAPDoc Label="ObjectElements:intro">
## An element of an object <M>M</M> is internally represented by a morphism from the <Q>structure object</Q>
## to the object <M>M</M>. In particular, the data structure for object elements
## automatically profits from the intrinsic realization of morphisms in the &homalg; project.
## <#/GAPDoc>
####################################
#
# representations:
#
####################################
## <#GAPDoc Label="IsElementOfAnObjectGivenByAMorphismRep">
## <ManSection>
## <Filt Type="Representation" Arg="M" Name="IsElementOfAnObjectGivenByAMorphismRep"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; representation of elements of finitley presented objects. <P/>
## (It is a representation of the &GAP; category <Ref Filt="IsHomalgElement"/>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsElementOfAnObjectGivenByAMorphismRep",
IsHomalgElement,
[ ] );
####################################
#
# immediate methods for properties:
#
####################################
##
InstallImmediateMethodToPullPropertiesOrAttributes(
IsElementOfAnObjectGivenByAMorphismRep,
IsElementOfAnObjectGivenByAMorphismRep,
[ "IsZero", [ "IsCyclicGenerator", "IsEpimorphism" ] ],
[ "IsZero", "IsCyclicGenerator" ],
UnderlyingMorphism );
##
InstallImmediateMethodToPushPropertiesOrAttributes( Twitter,
IsElementOfAnObjectGivenByAMorphismRep,
[ "IsZero", [ "IsCyclicGenerator", "IsEpimorphism" ] ],
UnderlyingMorphism );
####################################
#
# immediate methods for attributes:
#
####################################
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( DecideZero,
"for homalg elements",
[ IsHomalgElement ],
function( m )
## this will trigger DecideZero( UnderlyingMorphism( m ) )
IsZero( m );
return m;
end );
##
InstallMethod( ApplyMorphismToElement,
"for a morphism and an element",
[ IsHomalgStaticMorphism, IsHomalgElement ],
function( m, n )
if not IsIdenticalObj( Source( m ), SuperObject( n ) ) then
Error( "cannot apply morphism to element, element is not contained in source of morphism\n" );
fi;
return HomalgElement( PreCompose( UnderlyingMorphism( n ), m ) );
end );
##
InstallMethod( LT,
"for two homalg elements",
[ IsHomalgElement, IsHomalgElement ],
function( m, n )
## String must call DecideZero
return String( m ) < String( n );
end );
##
InstallMethod( \=,
"for two homalg elements",
[ IsHomalgElement, IsHomalgElement ],
function( m, n )
return UnderlyingMorphism( m ) = UnderlyingMorphism( n );
end );
##
InstallMethod( \+,
"for two homalg elements",
[ IsHomalgElement, IsHomalgElement ],
function( m, n )
return HomalgElement( UnderlyingMorphism( m ) + UnderlyingMorphism( n ) );
end );
##
InstallMethod( \+,
"for homalg ring elements",
[ IS_RAT, IsHomalgElement ],
function( a, b )
if IsZero( a ) then
return b;
elif HasIsZero( b ) and IsZero( b ) then
return a * One( b );
fi;
return a * One( b ) + b;
end );
##
InstallMethod( \+,
"for homalg ring elements",
[ IsHomalgElement, IS_RAT ],
function( a, b )
if IsZero( b ) then
return a;
elif HasIsZero( a ) and IsZero( a ) then
return b * One( a );
fi;
return a + b * One( a );
end );
##
InstallMethod( \+,
"for homalg elements",
[ IsHomalgElement and IsZero, IsHomalgElement ],
function( m, n )
return n;
end );
##
InstallMethod( \+,
"for homalg elements",
[ IsHomalgElement, IsHomalgElement and IsZero ],
function( m, n )
return m;
end );
##
InstallMethod( \-,
"for two homalg elements",
[ IsHomalgElement, IsHomalgElement ],
function( m, n )
return HomalgElement( UnderlyingMorphism( m ) - UnderlyingMorphism( n ) );
end );
##
InstallMethod( AdditiveInverseMutable,
"for homalg elements",
[ IsHomalgElement ],
function( m )
return HomalgElement( -UnderlyingMorphism( m ) );
end );
##
InstallMethod( \*,
"for homalg elements",
[ IsInt, IsHomalgElement ],
function( a, m )
return HomalgElement( a * UnderlyingMorphism( m ) );
end );
##
InstallMethod( ZeroMutable,
"for homalg elements",
[ IsHomalgElement ],
function( m )
return TheZeroElement( SuperObject( m ) );
end );
## if everything else fails
InstallMethod( \in,
"for homalg elements",
[ IsHomalgElement, IsStaticFinitelyPresentedObjectRep ],
function( m, M )
local phi;
phi := UnderlyingMorphism( m );
return IsIdenticalObj( Range( phi ), M );
end );
##
InstallMethod( \in,
"for homalg elements",
[ IsHomalgElement, IsStaticFinitelyPresentedObjectRep and HasUnderlyingSubobject ],
function( m, M )
return m in UnderlyingSubobject( M );
end );
## <#GAPDoc Label="in:elements">
## <ManSection>
## <Attr Arg="m, N" Name="in" Label="for elements"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Is the element <A>m</A> of the object <M>M</M> included in the subobject <A>N</A><M>\leq M</M>,
## i.e., does the morphism (with the unit object as source and <M>M</M> as target)
## underling the element <A>m</A> of <M>M</M> factor over the subobject morphism <A>N</A><M>\to M</M>?
## <P/>
## <#Include Label="HomalgElement_in:example">
## <P/>
## <Listing Type="Code"><![CDATA[
InstallMethod( \in,
"for homalg elements",
[ IsHomalgElement, IsStaticFinitelyPresentedSubobjectRep ],
function( m, N )
local phi, psi;
phi := UnderlyingMorphism( m );
psi := MorphismHavingSubobjectAsItsImage( N );
if not IsIdenticalObj( Range( phi ), Range( psi ) ) then
Error( "the super object of the subobject and the range ",
"of the morphism underlying the element do not coincide\n" );
fi;
return IsZero( PreCompose( phi, CokernelEpi( psi ) ) );
end );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \*,
"for a homalg element and a homalg morphism",
[ IsHomalgElement, IsHomalgMorphism and IsHomalgLeftObjectOrMorphismOfLeftObjects ],
function( m, phi )
return HomalgElement( UnderlyingMorphism( m ) * phi );
end );
##
InstallMethod( \*,
"for a homalg element and a homalg morphism",
[ IsHomalgMorphism and IsHomalgRightObjectOrMorphismOfRightObjects, IsHomalgElement ],
function( phi, m )
return HomalgElement( phi, UnderlyingMorphism( m ) );
end );
####################################
#
# constructor functions and methods:
#
####################################
##
InstallMethod( HomalgElement,
"for homalg static morphisms",
[ IsHomalgStaticMorphism ],
function( m )
local S, type, e;
S := Source( m );
type := HomalgCategory( S )!.TypeOfElements;
e := rec( );
if HasIsZero( m ) then
## Objectify
ObjectifyWithAttributes(
e, type,
UnderlyingMorphism, m,
SuperObject, Range( m ),
IsZero, IsZero( m ) );
else
## Objectify
ObjectifyWithAttributes(
e, type,
UnderlyingMorphism, m,
SuperObject, Range( m ) );
fi;
return e;
end );
##
InstallMethod( TheZeroElement,
"for homalg static objects",
[ IsHomalgStaticObject ],
function( M )
return HomalgElement( TheZeroMorphism( StructureObject( M ), M ) );
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( ViewObj,
"for homalg elements",
[ IsHomalgElement ],
function( o )
if IsZero( o ) then
ViewObj( o );
return;
fi;
Print( "<An element in " );
ViewObj( SuperObject( o ) );
Print( ">" );
end );
##
InstallMethod( ViewObj,
"for homalg elements",
[ IsHomalgElement and IsZero ],
function( o )
Print( 0 );
end );
##
InstallMethod( Display,
"for homalg elements",
[ IsHomalgElement ],
function( o )
if IsZero( o ) then
Display( o );
return;
fi;
Display( UnderlyingMorphism( o ) );
end );
##
InstallMethod( Display,
"for homalg elements",
[ IsHomalgElement and IsZero ],
function( o )
Print( 0, "\n" );
end );
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
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