Quelle HomalgMorphism.gd Sprache: unbekannt

# 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. 
## 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. 
## 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. 
## </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. 
## </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. 
## (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. 
## (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 ] );

Quelle HomalgMorphism.gd Sprache: unbekannt

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