SerreQuotientsBySpans.gd

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##                                               CAP package
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##  Copyright 2014, Sebastian Gutsche, TU Kaiserslautern
##                  Sebastian Posur,   RWTH Aachen
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#! @Chapter Serre Quotients
#! @Section Serre Quotients by Spans
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DeclareCategory( "IsSerreQuotientCategoryBySpansObject",
                 IsSerreQuotientCategoryObject );

DeclareCategory( "IsSerreQuotientCategoryBySpansMorphism",
                 IsSerreQuotientCategoryMorphism );

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## Attributes for serre category
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DeclareFilter( "WasCreatedAsSerreQuotientCategoryBySpans" );

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## Attributes
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## Constructor
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#! @BeginGroup
#! @Description
#!  Creates a Serre quotient category S with name <A>name</A> out of an Abelian category <A>A</A>.
#!  The Serre quotient category will be modeled upon the generalized morphisms by spans category of <A>A</A>
#!  If <A>name</A> is not given, a generic name is constructed out of the name of <A>A</A>.
#!  The argument <A>func</A> must be a unary function on the objects of <A>A</A> deciding the membership in
#!  the thick subcategory C mentioned above.
#! @Arguments A,func[,name]
#! @Returns a CAP category
DeclareOperation( "SerreQuotientCategoryBySpans",
                  [ IsCapCategory, IsFunction, IsString ] );

DeclareOperation( "SerreQuotientCategoryBySpans",
                  [ IsCapCategory, IsFunction ] );
#! @EndGroup

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and an object <A>M</A> in <A>A</A>,
#!  this constructor returns the corresponding object in the Serre quotient category.
#! @Arguments A/C, M
#! @Returns an object
DeclareOperation( "AsSerreQuotientCategoryBySpansObject",
                  [ IsCapCategory, IsCapCategoryObject ] );

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and a generalized morphism <A>phi</A> in
#!  the generalized morphism category <A>A/C</A> is modeled upon,
#!  this constructor returns the corresponding morphism in the Serre quotient category.
#! @Arguments A/C, phi
#! @Returns a morphism
DeclareOperation( "SerreQuotientCategoryBySpansMorphism",
                  [ IsCapCategory, IsGeneralizedMorphismBySpan ] );

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and three morphisms $\iota: M' \rightarrow M$,
#!  $\phi: M' \rightarrow N'$ and $\pi: N \rightarrow N'$ this operation contructs a
#!  morphism in the Serre quotient category.
#! @Arguments A/C, iota, phi, pi
#! @Returns a morphism
DeclareOperation( "SerreQuotientCategoryBySpansMorphism",
                  [ IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ] );

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and two morphisms $\alpha: M \rightarrow X$
#!  and $\beta: X \rightarrow N$
#!  this operation constructs the corresponding morphism in the Serre quotient category.
#! @Arguments A/C, alpha, beta
#! @Returns a morphism
DeclareOperation( "SerreQuotientCategoryBySpansMorphism",
                  [ IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism ] );

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and two morphisms $\alpha: X \rightarrow M$
#!  and $\beta: X \rightarrow N$
#!  this operation constructs the corresponding morphism in the Serre quotient category.
#! @Arguments A/C, alpha, beta
#! @Returns a morphism
DeclareOperation( "SerreQuotientCategoryBySpansMorphismWithRangeAid",
                  [ IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism ] );

#! @Description
#!  Given a Serre quotient category <A>A/C</A> modeled by spans and a morphism <A>phi</A> in <A>A</A>,
#!  this constructor returns the corresponding morphism in the Serre quotient category.
#! @Arguments A/C, phi
#! @Returns a morphism
DeclareOperation( "AsSerreQuotientCategoryBySpansMorphism",
                  [ IsCapCategory, IsCapCategoryMorphism ] );