Quelle HomalgSubobject.gi Sprache: unbekannt

# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#

##  Implementations for subobjects of objects of (Abelian) categories.

####################################
#
# representations:
#
####################################

##  <#GAPDoc Label="IsStaticFinitelyPresentedSubobjectRep">
##  <ManSection>
##    <Filt Type="Representation" Arg="M" Name="IsStaticFinitelyPresentedSubobjectRep"/>
##    <Returns><C>true</C> or <C>false</C></Returns>
##    <Description>
##      The &GAP; representation of finitley presented &homalg; subobjects of static objects. <P/>
##      (It is a representation of the &GAP; category <Ref Filt="IsHomalgStaticObject"/>,
##       which is a subrepresentation of the &GAP; representations
##       <C>IsStaticFinitelyPresentedObjectOrSubobjectRep</C> and
##       <C>IsFinitelyPresentedObjectRep</C>.)
##    <Listing Type="Code"><![CDATA[
DeclareRepresentation( "IsStaticFinitelyPresentedSubobjectRep",
        IsStaticFinitelyPresentedObjectOrSubobjectRep and
        IsFinitelyPresentedObjectRep,
        [ ] );
##  ]]></Listing>
##    </Description>
##  </ManSection>
##  <#/GAPDoc>

####################################
#
# methods for attributes:
#
####################################

##
InstallMethod( SuperObject,
        "for homalg subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep ],

  function( M )

    return Range( MorphismHavingSubobjectAsItsImage( M ) );

end );

####################################
#
# methods for operations:
#
####################################

##
InstallMethod( MorphismHavingSubobjectAsItsImage,
        "for homalg subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep ],
  function( M )

    if HasEmbeddingInSuperObject( M ) then
        return EmbeddingInSuperObject( M );
    fi;

    return M!.map_having_subobject_as_its_image;

end );

##
InstallMethod( HomalgCategory,
        "for homalg static subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep ],

  function( M )

    return HomalgCategory( SuperObject( M ) );

end );

##  <#GAPDoc Label="UnderlyingObject">
##  <ManSection>
##    <Oper Arg="M" Name="UnderlyingObject" Label="for subobjects"/>
##    <Returns>a &homalg; object</Returns>
##    <Description>
##      In case <A>M</A> was defined as a subobject of some object <M>L</M> the object underlying the subobject <M>M</M> is returned.
##    </Description>
##  </ManSection>
##  <#/GAPDoc>
##
InstallMethod( UnderlyingObject,
        "for homalg subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep ],

  function( M )

    return Source( EmbeddingInSuperObject( M ) );

end );

##
InstallMethod( \=,
        "for homalg subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],

  function( J, K )

    return IsSubset( J, K ) and IsSubset( K, J );

end );

##
InstallMethod( ByASmallerPresentation,
        "for homalg subobjects",
        [ IsStaticFinitelyPresentedSubobjectRep ],

  function( M )
    local emb;

    emb := EmbeddingInSuperObject( M );

    ByASmallerPresentation( Source( emb ) );

    DecideZero( emb );

    return M;

end );

##
InstallMethod( MatchPropertiesAndAttributesOfSubobjectAndUnderlyingObject,
        "for a subobject and its underlying object",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedObjectRep ],

  function( I, M )

    ## we don't check if M is the underlying object of I
    ## to avoid infinite loops as EmbeddingInSuperObject
    ## will be invoked
    if ConstructedAsAnIdeal( I ) then

        MatchPropertiesAndAttributes( I, M,
                LIOBJ.intrinsic_properties_shared_with_subobjects_and_ideals,
                LIOBJ.intrinsic_attributes_shared_with_subobjects_and_ideals );

    else

        MatchPropertiesAndAttributes( I, M,
                LIOBJ.intrinsic_properties_shared_with_subobjects_which_are_not_ideals,
                LIOBJ.intrinsic_attributes_shared_with_subobjects_which_are_not_ideals );

    fi;

end );

##
InstallMethod( \=,
        "for a homalg subobject and a homalg structure object",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStructureObject ],

  function( J, R )
    local equal;

    if not IsIdenticalObj( StructureObject( J ), R ) then
        Error( "the structure object of the subobject and the given structure object are not identical\n" );
    fi;

    equal := IsSubset( J, FullSubobject( SuperObject( J ) ) );

    if equal then
        SetIsZero( J, IsZero( R ) );
    fi;

    return equal;

end );

##
InstallMethod( \=,
        "for a homalg structure object and a homalg subobject",
        [ IsStructureObject, IsStaticFinitelyPresentedSubobjectRep ],

  function( R, J )

    return J = R;

end );

##
InstallMethod( IsOne,
        "for a homalg subobject and a homalg structure object",
        [ IsStaticFinitelyPresentedSubobjectRep ],

  function( J )

    ## the following uses IsSubset in one direction only; see above
    return J = StructureObject( J );

end );

##
InstallMethod( IsSubset,
        "for a homalg subobject and a homalg structure object",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStructureObject ],

  function( J, R )

    ## the following uses IsSubset in one direction only; see above
    return J = R;

end );

####################################
#
# constructor functions and methods:
#
####################################

##
InstallMethod( \*,
        "for homalg subobjects",
        [ IsStructureObject, IsStaticFinitelyPresentedSubobjectRep ], 20001,

  function( R, M )

    return ImageSubobject( R * MorphismHavingSubobjectAsItsImage( M ) );

end );

##  <#GAPDoc Label="Subobject">
##  <ManSection>
##    <Oper Arg="phi" Name="Subobject" Label="constructor for subobjects using morphisms"/>
##    <Returns>a &homalg; subobject</Returns>
##    <Description>
##      A synonym of <Ref Attr="ImageSubobject"/>.
##    </Description>
##  </ManSection>
##  <#/GAPDoc>
##
InstallMethod( Subobject,
        "constructor for homalg subobjects",
        [ IsHomalgStaticMorphism ],

  ImageSubobject );

##
InstallMethod( Pullback,
        "for a morphism of structure objects and a static subobject",
        [ IsStructureObjectMorphism, IsStaticFinitelyPresentedSubobjectRep ],

  function( phi, I )
    local mor;

    mor := MorphismHavingSubobjectAsItsImage( I );

    mor := Pullback( phi, mor );

    return ImageSubobject( mor );

end );

Quelle HomalgSubobject.gi Sprache: unbekannt

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