GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>
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<!-- This is an automatically generated file. -->
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<Chapter Label="Chapter_Serre_Quotients">
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<Heading>Serre Quotients</Heading>
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Serre quotients are implemented using generalized morphisms. A Serre quotient category
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is the quotient of an abelian category A by a thick subcategory C. The objects of the quotient
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are the objects from A, the morphisms are a limit construction. In the implementation
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those morphisms are modeled by generalized morphisms, and therefore there are,
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like in the generalized morphism case, three types of Serre quotients.
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<Section Label="Chapter_Serre_Quotients_Section_General_operations">
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<Heading>General operations</Heading>
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As in the generalized morphism case, the generic constructors depend on the
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generalized morphism standard. Please note that for implementations the specialized
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constructors should be used.
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<ManSection>
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  <Filt Arg="arg" Name="IsSerreQuotientCategoryObject" Label="for IsCapCategoryObject"/>
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 <Returns><C>true</C> or <C>false</C>
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</Returns>
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 <Description>
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The category of objects in the category of Serre quotients.
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For actual objects this needs to be specialized.
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 </Description>
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</ManSection>
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<ManSection>
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  <Filt Arg="arg" Name="IsSerreQuotientCategoryMorphism" Label="for IsCapCategoryMorphism"/>
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 <Returns><C>true</C> or <C>false</C>
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</Returns>
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 <Description>
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The category of morphisms in the category of Serre quotients.
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For actual morphisms this needs to be specialized.
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 </Description>
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</ManSection>
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<ManSection Label="AutoDoc_generated_group1">
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  <Oper Arg="A,func[,name]" Name="SerreQuotientCategory" Label="for IsCapCategory, IsFunction, IsString"/>
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 <Returns>a CAP category
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</Returns>
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 <Description>
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Creates a Serre quotient category <A>S</A> with name <A>name</A> out of an Abelian category <A>A</A>.
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If <A>name</A> is not given, a generic name is constructed out of the name of <A>A</A>.
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The argument <A>func</A> must be a unary function on the objects of <A>A</A> deciding the membership in
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the thick subcategory C mentioned above.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, M" Name="AsSerreQuotientCategoryObject" Label="for IsCapCategory, IsCapCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and an object <A>M</A> in <A>A</A>,
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this constructor returns the corresponding object in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="SerreQuotientCategoryMorphism" Label="for IsCapCategory, IsGeneralizedMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and a generalized morphism <A>phi</A> in
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the generalized morphism category <A>A/C</A> is modeled upon,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, iota, phi, pi" Name="SerreQuotientCategoryMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and three morphisms <Math>\iota: M' \rightarrow M</Math>,
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<Math>\phi: M' \rightarrow N'</Math> and <Math>\pi: N \rightarrow N'</Math> this operation contructs a
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morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and two morphisms of the form <Math>\alpha: X \rightarrow M</Math>
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and <Math>\beta: X \rightarrow N</Math> or <Math>\alpha: M \rightarrow X</Math> and <Math>\beta: N \rightarrow X</Math>,
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this operation constructs the corresponding morphism in the Serre quotient category.
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This operation is only implemented if <A>A/C</A> is
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modeled upon a span generalized morphism category in the first option or upon a cospan
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category in the second.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryMorphismWithSourceAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and two morphisms <Math>\alpha: M \rightarrow X</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryMorphismWithRangeAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and two morphisms <Math>\alpha: X \rightarrow M</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="AsSerreQuotientCategoryMorphism" Label="for IsCapCategory, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> and a morphism <A>phi</A> in <A>A</A>,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="C" Name="SubcategoryMembershipTestFunctionForSerreQuotient" Label="for IsCapCategory"/>
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 <Returns>a function
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</Returns>
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 <Description>
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When a Serre quotient category is created, a membership function for
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the subcategory is given. This attribute stores and returns this function
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="A/C" Name="UnderlyingHonestCategory" Label="for IsCapCategory"/>
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 <Returns>a category
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</Returns>
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 <Description>
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For a Serre quotient category <A>A/C</A> this attribute returns the category <A>A</A>.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="A/C" Name="UnderlyingGeneralizedMorphismCategory" Label="for IsCapCategory"/>
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 <Returns>a category
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</Returns>
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 <Description>
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For a Serre quotient category <A>A/C</A> this attribute returns generalized morphism category the quotient is modelled upon.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="M" Name="UnderlyingGeneralizedObject" Label="for IsSerreQuotientCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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For an object <A>M</A> in the Serre quotient category A/C this attribute returns the
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corresponding object in the generalized morphism category the quotient is modelled upon.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="M" Name="UnderlyingHonestObject" Label="for IsSerreQuotientCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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For an object <A>M</A> in the Serre quotient category A/C this attribute returns the
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corresponding object in <A>A</A>.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="phi" Name="UnderlyingGeneralizedMorphism" Label="for IsSerreQuotientCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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For a morphism <A>phi</A> in the Serre quotient category A/C this attribute returns the
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corresponding generalized morphism in the generalized morphism category the quotient is modelled upon.
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 </Description>
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</ManSection>
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<ManSection>
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  <Attr Arg="A/C" Name="CanonicalProjection" Label="for IsCapCategory"/>
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 <Returns>a functor
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A>, this operation returns the canonical projection functor
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<Math> A \rightarrow A/C </Math>.
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 </Description>
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</ManSection>
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</Section>
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<Section Label="Chapter_Serre_Quotients_Section_Serre_quotients_by_cospans">
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<Heading>Serre quotients by cospans</Heading>
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<ManSection Label="AutoDoc_generated_group2">
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  <Oper Arg="A,func[,name]" Name="SerreQuotientCategoryByCospans" Label="for IsCapCategory, IsFunction, IsString"/>
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 <Returns>a CAP category
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</Returns>
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 <Description>
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Creates a Serre quotient category S with name <A>name</A> out of an Abelian category <A>A</A>.
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The Serre quotient category will be modeled upon the generalized morphisms by cospans category of <A>A</A>
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If <A>name</A> is not given, a generic name is constructed out of the name of <A>A</A>.
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The argument <A>func</A> must be a unary function on the objects of <A>A</A> deciding the membership in
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the thick subcategory C mentioned above.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, M" Name="AsSerreQuotientCategoryByCospansObject" Label="for IsCapCategory, IsCapCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and an object <A>M</A> in <A>A</A>,
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this constructor returns the corresponding object in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="SerreQuotientCategoryByCospansMorphism" Label="for IsCapCategory, IsGeneralizedMorphismByCospan"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and a generalized morphism <A>phi</A> in
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the generalized morphism category <A>A/C</A> is modeled upon,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, iota, phi, pi" Name="SerreQuotientCategoryByCospansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and three morphisms <Math>\iota: M' \rightarrow M</Math>,
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<Math>\phi: M' \rightarrow N'</Math> and <Math>\pi: N \rightarrow N'</Math> this operation contructs a
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morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryByCospansMorphismWithSourceAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and two morphisms <Math>\alpha: M \rightarrow X</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryByCospansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and two morphisms <Math>\alpha: X \rightarrow M</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="AsSerreQuotientCategoryByCospansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by cospans and a morphism <A>phi</A> in <A>A</A>,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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</Section>
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<Section Label="Chapter_Serre_Quotients_Section_Serre_Quotients_by_Spans">
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<Heading>Serre Quotients by Spans</Heading>
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<ManSection Label="AutoDoc_generated_group3">
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  <Oper Arg="A,func[,name]" Name="SerreQuotientCategoryBySpans" Label="for IsCapCategory, IsFunction, IsString"/>
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 <Returns>a CAP category
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</Returns>
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 <Description>
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Creates a Serre quotient category S with name <A>name</A> out of an Abelian category <A>A</A>.
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The Serre quotient category will be modeled upon the generalized morphisms by spans category of <A>A</A>
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If <A>name</A> is not given, a generic name is constructed out of the name of <A>A</A>.
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The argument <A>func</A> must be a unary function on the objects of <A>A</A> deciding the membership in
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the thick subcategory C mentioned above.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, M" Name="AsSerreQuotientCategoryBySpansObject" Label="for IsCapCategory, IsCapCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and an object <A>M</A> in <A>A</A>,
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this constructor returns the corresponding object in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="SerreQuotientCategoryBySpansMorphism" Label="for IsCapCategory, IsGeneralizedMorphismBySpan"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and a generalized morphism <A>phi</A> in
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the generalized morphism category <A>A/C</A> is modeled upon,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, iota, phi, pi" Name="SerreQuotientCategoryBySpansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and three morphisms <Math>\iota: M' \rightarrow M</Math>,
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<Math>\phi: M' \rightarrow N'</Math> and <Math>\pi: N \rightarrow N'</Math> this operation contructs a
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morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryBySpansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and two morphisms <Math>\alpha: M \rightarrow X</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryBySpansMorphismWithRangeAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and two morphisms <Math>\alpha: X \rightarrow M</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="AsSerreQuotientCategoryBySpansMorphism" Label="for IsCapCategory, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by spans and a morphism <A>phi</A> in <A>A</A>,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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</Section>
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<Section Label="Chapter_Serre_Quotients_Section_Serre_Quotients_modeled_by_three_arrows">
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<Heading>Serre Quotients modeled by three arrows</Heading>
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<ManSection Label="AutoDoc_generated_group4">
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  <Oper Arg="A,func[,name]" Name="SerreQuotientCategoryByThreeArrows" Label="for IsCapCategory, IsFunction, IsString"/>
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 <Returns>a CAP category
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</Returns>
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 <Description>
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Creates a Serre quotient category S with name <A>name</A> out of an Abelian category <A>A</A>.
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The Serre quotient category will be modeled upon the generalized morphisms by three arrows category of <A>A</A>
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If <A>name</A> is not given, a generic name is constructed out of the name of <A>A</A>.
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The argument <A>func</A> must be a unary function on the objects of <A>A</A> deciding the membership in
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the thick subcategory C mentioned above.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, M" Name="AsSerreQuotientCategoryByThreeArrowsObject" Label="for IsCapCategory, IsCapCategoryObject"/>
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 <Returns>an object
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and an object <A>M</A> in <A>A</A>,
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this constructor returns the corresponding object in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="SerreQuotientCategoryByThreeArrowsMorphism" Label="for IsCapCategory, IsGeneralizedMorphismByThreeArrows"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and a generalized morphism <A>phi</A> in
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the generalized morphism category <A>A/C</A> is modeled upon,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, iota, phi, pi" Name="SerreQuotientCategoryByThreeArrowsMorphism" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and three morphisms <Math>\iota: M' \rightarrow M</Math>,
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<Math>\phi: M' \rightarrow N'</Math> and <Math>\pi: N \rightarrow N'</Math> this operation contructs a
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morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and two morphisms <Math>\alpha: M \rightarrow X</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, alpha, beta" Name="SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid" Label="for IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and two morphisms <Math>\alpha: X \rightarrow M</Math>
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and <Math>\beta: X \rightarrow N</Math>
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this operation constructs the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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<ManSection>
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  <Oper Arg="A/C, phi" Name="AsSerreQuotientCategoryByThreeArrowsMorphism" Label="for IsCapCategory, IsCapCategoryMorphism"/>
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 <Returns>a morphism
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</Returns>
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 <Description>
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Given a Serre quotient category <A>A/C</A> modeled by three arrows and a morphism <A>phi</A> in <A>A</A>,
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this constructor returns the corresponding morphism in the Serre quotient category.
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 </Description>
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</ManSection>
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</Section>
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</Chapter>
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