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# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Declarations
#
## Declaration stuff for homalg complexes.
####################################
#
# categories:
#
####################################
# a new GAP-category:
## <#GAPDoc Label="IsHomalgComplex">
## <ManSection>
## <Filt Type="Category" Arg="C" Name="IsHomalgComplex"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of &homalg; (co)complexes. <P/>
## (It is a subcategory of the &GAP; category <C>IsHomalgObject</C>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsHomalgComplex",
IsHomalgObject );
####################################
#
# properties:
#
####################################
## <#GAPDoc Label="IsSequence">
## <ManSection>
## <Prop Arg="C" Name="IsSequence"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if all maps in <A>C</A> are well-defined.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSequence",
IsHomalgComplex );
## <#GAPDoc Label="IsComplex">
## <ManSection>
## <Prop Arg="C" Name="IsComplex"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if <A>C</A> is complex.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsComplex",
IsHomalgComplex );
## <#GAPDoc Label="IsAcyclic">
## <ManSection>
## <Prop Arg="C" Name="IsAcyclic"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is acyclic, i.e. exact except at its boundaries.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsAcyclic",
IsHomalgComplex );
## <#GAPDoc Label="IsRightAcyclic">
## <ManSection>
## <Prop Arg="C" Name="IsRightAcyclic"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is acyclic, i.e. exact except at its left boundary.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRightAcyclic",
IsHomalgComplex );
## <#GAPDoc Label="IsLeftAcyclic">
## <ManSection>
## <Prop Arg="C" Name="IsLeftAcyclic"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is acyclic, i.e. exact except at its right boundary.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLeftAcyclic",
IsHomalgComplex );
## <#GAPDoc Label="IsGradedObject">
## <ManSection>
## <Prop Arg="C" Name="IsGradedObject"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is a graded object, i.e. if all maps between the objects in <A>C</A> vanish.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsGradedObject",
IsHomalgComplex );
## <#GAPDoc Label="IsExactSequence">
## <ManSection>
## <Prop Arg="C" Name="IsExactSequence"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is exact.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsExactSequence",
IsHomalgComplex );
## <#GAPDoc Label="IsShortExactSequence">
## <ManSection>
## <Prop Arg="C" Name="IsShortExactSequence"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is a short exact sequence.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsShortExactSequence", ## we also need this as property!!!
IsHomalgComplex );
## <#GAPDoc Label="IsSplitShortExactSequence">
## <ManSection>
## <Prop Arg="C" Name="IsSplitShortExactSequence"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is a split short exact sequence.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSplitShortExactSequence",
IsHomalgComplex );
## <#GAPDoc Label="IsTriangle">
## <ManSection>
## <Prop Arg="C" Name="IsTriangle"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Set to true if the &homalg; complex <A>C</A> is a triangle.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsTriangle",
IsHomalgComplex );
## <#GAPDoc Label="IsExactTriangle">
## <ManSection>
## <Prop Arg="C" Name="IsExactTriangle"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; complex <A>C</A> is an exact triangle.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsExactTriangle",
IsHomalgComplex );
DeclareProperty( "IsATwoSequence", ## the output of AsATwoSequence (and only this) is marked as IsATwoSequence in order to distinguish
IsHomalgComplex ); ## between different methods for DefectOfExactness which all apply to complexes
####################################
#
# attributes:
#
####################################
## <#GAPDoc Label="BettiTable:complex">
## <ManSection>
## <Attr Arg="C" Name="BettiTable" Label="for complexes"/>
## <Returns>a &homalg; diagram</Returns>
## <Description>
## The Betti diagram of the &homalg; complex <A>C</A> of graded modules.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BettiTable",
IsHomalgComplex );
DeclareSynonym( "BettiDiagram", BettiTable );
## <#GAPDoc Label="FiltrationByShortExactSequence">
## <ManSection>
## <Attr Arg="C" Name="FiltrationByShortExactSequence" Label="for complexes"/>
## <Returns>a &homalg; diagram</Returns>
## <Description>
## The filtration induced by the short exact sequence <A>C</A> on its middle object.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "FiltrationByShortExactSequence",
IsHomalgComplex );
####################################
#
# global functions and operations:
#
####################################
# constructors:
DeclareGlobalFunction( "HomalgComplex" );
DeclareGlobalFunction( "HomalgCocomplex" );
DeclareOperation( "*",
[ IsStructureObject, IsHomalgComplex ] );
DeclareOperation( "*",
[ IsHomalgComplex, IsStructureObject ] );
# basic operations:
DeclareOperation( "ObjectDegreesOfComplex",
[ IsHomalgComplex ] );
DeclareOperation( "MorphismDegreesOfComplex",
[ IsHomalgComplex ] );
DeclareOperation( "CertainMorphism",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "CertainObject",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "MorphismsOfComplex",
[ IsHomalgComplex ] );
DeclareOperation( "ObjectsOfComplex",
[ IsHomalgComplex ] );
DeclareOperation( "LowestDegree",
[ IsHomalgComplex ] );
DeclareOperation( "HighestDegree",
[ IsHomalgComplex ] );
DeclareOperation( "LowestDegreeObject",
[ IsHomalgComplex ] );
DeclareOperation( "HighestDegreeObject",
[ IsHomalgComplex ] );
DeclareOperation( "LowestDegreeMorphism",
[ IsHomalgComplex ] );
DeclareOperation( "HighestDegreeMorphism",
[ IsHomalgComplex ] );
DeclareOperation( "SupportOfComplex",
[ IsHomalgComplex ] );
DeclareOperation( "Add",
[ IsHomalgComplex, IsHomalgMorphism ] );
DeclareOperation( "Add",
[ IsHomalgMorphism, IsHomalgComplex ] );
DeclareOperation( "Add",
[ IsHomalgComplex, IsHomalgStaticObject ] );
DeclareOperation( "Add",
[ IsHomalgStaticObject, IsHomalgComplex ] );
DeclareOperation( "Shift",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "Reflect",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "Reflect",
[ IsHomalgComplex ] );
DeclareOperation( "Subcomplex",
[ IsHomalgComplex, IsInt, IsInt ] );
DeclareOperation( "CertainMorphismAsSubcomplex",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "CertainTwoMorphismsAsSubcomplex",
[ IsHomalgComplex, IsInt ] );
DeclareOperation( "PreCompose",
[ IsHomalgComplex, IsHomalgComplex ] );
DeclareOperation( "LongSequence",
[ IsHomalgComplex ] );
DeclareOperation( "SetCurrentResolution",
[ IsHomalgStaticObject, IsHomalgComplex ] );
[ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
]
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