|
# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Declarations
#
## Declarations for morphisms of (Abelian) categories.
## <#GAPDoc Label="IsHomalgMorphism">
## <ManSection>
## <Filt Type="Category" Arg="phi" Name="IsHomalgMorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## This is the super &GAP;-category which will include the &GAP;-categories
## <Ref Filt="IsHomalgStaticMorphism"/> and <Ref Filt="IsHomalgChainMorphism"/>.
## We need this &GAP;-category to be able to build complexes with *objects*
## being objects of &homalg; categories or again complexes.
## We need this GAP-category to be able to build chain morphisms with *morphisms*
## being morphisms of &homalg; categories or again chain morphisms. <Br/>
## CAUTION: Never let &homalg; morphisms (which are not endomorphisms)
## be multiplicative elements!!
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgMorphism",
IsHomalgStaticObjectOrMorphism and
IsAdditiveElementWithInverse );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## <#GAPDoc Label="IsHomalgStaticMorphism">
## <ManSection>
## <Filt Type="Category" Arg="phi" Name="IsHomalgStaticMorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## This is the super &GAP;-category which will include the &GAP;-categories
## <C>IsHomalgMap</C>, etc. <Br/>
## CAUTION: Never let homalg morphisms (which are not endomorphisms)
## be multiplicative elements!!
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgStaticMorphism",
IsHomalgMorphism );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## <#GAPDoc Label="IsHomalgEndomorphism">
## <ManSection>
## <Filt Type="Category" Arg="phi" Name="IsHomalgEndomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## This is the super &GAP;-category which will include the &GAP;-categories
## <C>IsHomalgSelfMap</C>, <Ref Filt="IsHomalgChainEndomorphism"/>, etc.
## be multiplicative elements!!
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgEndomorphism",
IsHomalgMorphism and
IsMultiplicativeElementWithInverse );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
####################################
#
# properties:
#
####################################
## <#GAPDoc Label="IsMorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsMorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## <C>IsMorphism</C>=<C>true</C> means one of the following:
## <List>
## <Item>
## The property method <C>IsMorphism</C>(<A>phi</A>) was explicitly invoked by the user
## and it returned <C>true</C>, where prior to the invocation <C>HasIsMorphism</C>(<A>phi</A>) was <C>false</C>.
## The method is meant to check the integrity of the data structure at the time of it invocation.
## What this precisely means depends on the specific &homalg;-based package.
## </Item>
## <Item>
## The user has explicitly <C>SetIsMorphism</C>(<A>phi</A>, <C>true</C>).
## </Item>
## <Item>
## The morphism <A>phi</A> is output of a categorical procedure where <C>IsMorphism</C>
## has become <C>true</C> for all morphisms in the input.
## </Item>
## <Item>
## The morphism <A>phi</A> is output of a categorical procedure which gurantees the
## integrity of the data structure of its output independent of its input.
## </Item>
## </List>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsMorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsGeneralizedMorphismWithFullDomain">
## <ManSection>
## <Prop Arg="phi" Name="IsGeneralizedMorphismWithFullDomain"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if <A>phi</A> is a generalized morphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsGeneralizedMorphismWithFullDomain",
IsHomalgMorphism );
## <#GAPDoc Label="IsGeneralizedEpimorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsGeneralizedEpimorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if <A>phi</A> is a generalized epimorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsGeneralizedEpimorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsGeneralizedMonomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsGeneralizedMonomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if <A>phi</A> is a generalized monomorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsGeneralizedMonomorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsGeneralizedIsomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsGeneralizedIsomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if <A>phi</A> is a generalized isomorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsGeneralizedIsomorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsOne">
## <ManSection>
## <Prop Arg="phi" Name="IsOne"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is the identity morphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsOne",
IsHomalgMorphism );
## <#GAPDoc Label="IsMonomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsMonomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is a monomorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsMonomorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsEpimorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsEpimorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is an epimorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsEpimorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsSplitMonomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsSplitMonomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is a split monomorphism. <Br/>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsSplitMonomorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsSplitEpimorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsSplitEpimorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is a split epimorphism. <Br/>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsSplitEpimorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsIsomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsIsomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is an isomorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsIsomorphism",
IsHomalgMorphism );
## <#GAPDoc Label="IsAutomorphism">
## <ManSection>
## <Prop Arg="phi" Name="IsAutomorphism"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is an automorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsAutomorphism", ## do not make an ``and''-filter out of this property (I hope the other GAP packages respect this)
IsHomalgMorphism );
## <#GAPDoc Label="IsIdempotent">
## <ManSection>
## <Prop Arg="phi" Name="IsIdempotent"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; morphism <A>phi</A> is an automorphism.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsIdempotent", ## do not make an ``and''-filter out of this property (I hope the other GAP packages respect this)
IsHomalgMorphism );
####################################
#
# attributes:
#
####################################
DeclareAttribute( "Genesis",
IsHomalgStaticMorphism, "mutable" );
## <#GAPDoc Label="Source">
## <ManSection>
## <Attr Arg="phi" Name="Source"/>
## <Returns>a &homalg; object</Returns>
## <Description>
## The source of the &homalg; morphism <A>phi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "Source",
IsHomalgMorphism );
## <#GAPDoc Label="Range">
## <ManSection>
## <Attr Arg="phi" Name="Range"/>
## <Returns>a &homalg; object</Returns>
## <Description>
## The target (range) of the &homalg; morphism <A>phi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "Range",
IsHomalgMorphism );
## <#GAPDoc Label="ImageSubobject">
## <ManSection>
## <Attr Arg="phi" Name="ImageSubobject"/>
## <Returns>a &homalg; subobject</Returns>
## <Description>
## This constructor returns the finitely generated image of the &homalg; morphism <A>phi</A>
## as a subobject of the &homalg; object <C>Range</C>(<A>phi</A>) with generators given by <A>phi</A>
## applied to the generators of its source object.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "ImageSubobject",
IsHomalgMorphism );
## <#GAPDoc Label="KernelSubobject">
## <ManSection>
## <Attr Arg="phi" Name="KernelSubobject"/>
## <Returns>a &homalg; subobject</Returns>
## <Description>
## This constructor returns the finitely generated kernel of the &homalg; morphism <A>phi</A>
## as a subobject of the &homalg; object <C>Source</C>(<A>phi</A>) with generators given by
## the syzygies of <A>phi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "KernelSubobject",
IsHomalgMorphism );
## <#GAPDoc Label="MorphismAid">
## <ManSection>
## <Attr Arg="phi" Name="MorphismAid"/>
## <Returns>a &homalg; morphism</Returns>
## <Description>
## The morphism aid map of a true generalized map. <Br/>
## (no method installed)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "GetMorphismAid",
[ IsHomalgMorphism ] );
DeclareAttributeWithCustomGetter( "MorphismAid",
IsHomalgMorphism,
GetMorphismAid );
## <#GAPDoc Label="InverseOfGeneralizedMorphismWithFullDomain">
## <ManSection>
## <Attr Arg="phi" Name="InverseOfGeneralizedMorphismWithFullDomain"/>
## <Returns>a &homalg; morphism</Returns>
## <Description>
## The generalized inverse of the epimorphism <A>phi</A> (cf. <Cite Key="BaSF" Where="Cor. 4.8"/>)).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InverseOfGeneralizedMorphismWithFullDomain",
IsHomalgMorphism );
## <#GAPDoc Label="DirectSummands">
## <ManSection>
## <Attr Arg="phi" Name="DirectSummands"/>
## <Returns>a list of &homalg; morphisms</Returns>
## <Description>
## In case <A>phi</A> was created using DiagonalMorphism
## is then set to the direct factor morphisms.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DirectSummands",
IsHomalgMorphism );
DeclareAttribute( "DegreeOfMorphism",
IsStructureObjectMorphism );
## <#GAPDoc Label="DegreeOfMorphism">
## <ManSection>
## <Attr Arg="phi" Name="DegreeOfMorphism"/>
## <Returns>an integer</Returns>
## <Description>
## The degree of the morphism <A>phi</A> between graded objects. <Br/>
## (no method installed)
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "DegreeOfMorphism",
IsHomalgMorphism );
####################################
#
# global functions and operations:
#
####################################
# constructors:
DeclareOperation( "RemoveMorphismAid",
[ IsHomalgMorphism ] );
DeclareOperation( "GeneralizedMorphism",
[ IsHomalgMorphism, IsObject ] );
DeclareOperation( "AddToMorphismAid",
[ IsHomalgMorphism, IsObject ] );
DeclareOperation( "AssociatedMorphismOfGeneralizedMorphismWithFullDomain",
[ IsHomalgMorphism ] );
DeclareOperation( "AnIsomorphism",
[ IsHomalgObject ] );
DeclareOperation( "Subobject",
[ IsHomalgMorphism ] );
DeclareOperation( "Pullback",
[ IsStructureObjectMorphism, IsHomalgMorphism ] );
DeclareOperation( "*",
[ IsStructureObject, IsHomalgMorphism ] );
DeclareOperation( "*",
[ IsHomalgMorphism, IsStructureObject ] );
DeclareGlobalFunction( "INSTALL_TODO_LIST_ENTRIES_FOR_MORPHISMS_AND_IMAGE_EMBEDDINGS" );
# basic operations:
DeclareOperation( "PairOfPositionsOfTheDefaultPresentations",
[ IsHomalgMorphism ] );
DeclareOperation( "AreComparableMorphisms",
[ IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "AreComposableMorphisms",
[ IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "*", ## this must remain, since an element in IsHomalgMorphism
[ IsHomalgMorphism, IsHomalgMorphism ] ); ## is not a priori IsMultiplicativeElement
DeclareOperation( "^", ## this must remain, since an element in IsHomalgMorphism
[ IsHomalgMorphism, IsInt ] ); ## is not a priori IsMultiplicativeElement
DeclareOperation( "PreInverse",
[ IsHomalgMorphism ] );
DeclareOperation( "PostInverse",
[ IsHomalgMorphism ] );
DeclareOperation( "CompleteImageSquare",
[ IsHomalgMorphism, IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "CompleteKernelSquare",
[ IsHomalgMorphism, IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "DiagonalMorphismOp",
[ IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "DiagonalMorphismOp",
[ IsList, IsHomalgMorphism ] );
DeclareGlobalFunction( "DiagonalMorphism" );
DeclareOperation( "UpdateObjectsByMorphism",
[ IsHomalgMorphism ] );
DeclareOperation( "SetPropertiesOfGeneralizedMorphism",
[ IsHomalgMorphism, IsHomalgMorphism ] );
DeclareOperation( "/",
[ IsHomalgMorphism, IsHomalgStaticObject ] );
[ Dauer der Verarbeitung: 0.34 Sekunden
(vorverarbeitet)
]
|
