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  7 Serre Quotients
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  Serre  quotients  are  implemented  using  generalized  morphisms.  A  Serre
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  quotient  category  is  the  quotient  of  an  abelian category A by a thick
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  subcategory  C.  The  objects  of  the  quotient are the objects from A, the
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  morphisms  are  a  limit construction. In the implementation those morphisms
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  are  modeled  by generalized morphisms, and therefore there are, like in the
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  generalized morphism case, three types of Serre quotients.
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  7.1 General operations
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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
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  specialized constructors should be used.
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  7.1-1 IsSerreQuotientCategoryObject
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  IsSerreQuotientCategoryObject( arg )  filter
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  Returns:  true or false
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  The  category  of  objects  in  the  category of Serre quotients. For actual
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  objects this needs to be specialized.
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  7.1-2 IsSerreQuotientCategoryMorphism
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  IsSerreQuotientCategoryMorphism( arg )  filter
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  Returns:  true or false
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  The  category  of  morphisms  in the category of Serre quotients. For actual
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  morphisms this needs to be specialized.
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  7.1-3 SerreQuotientCategory
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  SerreQuotientCategory( A, func[, name] )  operation
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  Returns:  a CAP category
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  Creates  a  Serre  quotient  category  S  with  name  name out of an Abelian
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  category  A.  If name is not given, a generic name is constructed out of the
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  name  of  A.  The argument func must be a unary function on the objects of A
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  deciding the membership in the thick subcategory C mentioned above.
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  7.1-4 AsSerreQuotientCategoryObject
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  AsSerreQuotientCategoryObject( A/C, M )  operation
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  Returns:  an object
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  Given  a  Serre quotient category A/C and an object M in A, this constructor
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  returns the corresponding object in the Serre quotient category.
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  7.1-5 SerreQuotientCategoryMorphism
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  SerreQuotientCategoryMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient category A/C and a generalized morphism phi in the
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  generalized  morphism category A/C is modeled upon, this constructor returns
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  the corresponding morphism in the Serre quotient category.
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  7.1-6 SerreQuotientCategoryMorphism
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  SerreQuotientCategoryMorphism( A/C, iota, phi, pi )  operation
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  Returns:  a morphism
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  Given   a  Serre  quotient  category  A/C  and  three  morphisms  \iota:  M'
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  \rightarrow  M,  \phi:  M'  \rightarrow  N'  and  \pi: N \rightarrow N' this
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  operation contructs a morphism in the Serre quotient category.
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  7.1-7 SerreQuotientCategoryMorphism
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  SerreQuotientCategoryMorphism( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a Serre quotient category A/C and two morphisms of the form \alpha: X
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  \rightarrow  M  and  \beta:  X  \rightarrow N or \alpha: M \rightarrow X and
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  \beta: N \rightarrow X, this operation constructs the corresponding morphism
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  in the Serre quotient category. This operation is only implemented if A/C is
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  modeled  upon  a  span  generalized morphism category in the first option or
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  upon a cospan category in the second.
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  7.1-8 SerreQuotientCategoryMorphismWithSourceAid
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  SerreQuotientCategoryMorphismWithSourceAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a Serre quotient category A/C and two morphisms \alpha: M \rightarrow
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  X  and  \beta:  X  \rightarrow N this operation constructs the corresponding
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  morphism in the Serre quotient category.
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  7.1-9 SerreQuotientCategoryMorphismWithRangeAid
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  SerreQuotientCategoryMorphismWithRangeAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a Serre quotient category A/C and two morphisms \alpha: X \rightarrow
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  M  and  \beta:  X  \rightarrow N this operation constructs the corresponding
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  morphism in the Serre quotient category.
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  7.1-10 AsSerreQuotientCategoryMorphism
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  AsSerreQuotientCategoryMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  and  a  morphism  phi  in  A, this
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  constructor  returns  the  corresponding  morphism  in  the  Serre  quotient
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  category.
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  7.1-11 SubcategoryMembershipTestFunctionForSerreQuotient
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  SubcategoryMembershipTestFunctionForSerreQuotient( C )  attribute
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  Returns:  a function
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  When  a  Serre  quotient  category is created, a membership function for the
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  subcategory is given. This attribute stores and returns this function
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  7.1-12 UnderlyingHonestCategory
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  UnderlyingHonestCategory( A/C )  attribute
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  Returns:  a category
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  For a Serre quotient category A/C this attribute returns the category A.
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  7.1-13 UnderlyingGeneralizedMorphismCategory
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  UnderlyingGeneralizedMorphismCategory( A/C )  attribute
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  Returns:  a category
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  For  a  Serre  quotient  category  A/C  this  attribute  returns generalized
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  morphism category the quotient is modelled upon.
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  7.1-14 UnderlyingGeneralizedObject
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  UnderlyingGeneralizedObject( M )  attribute
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  Returns:  an object
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  For  an  object  M in the Serre quotient category A/C this attribute returns
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  the  corresponding  object in the generalized morphism category the quotient
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  is modelled upon.
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  7.1-15 UnderlyingHonestObject
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  UnderlyingHonestObject( M )  attribute
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  Returns:  an object
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  For  an  object  M in the Serre quotient category A/C this attribute returns
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  the corresponding object in A.
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  7.1-16 UnderlyingGeneralizedMorphism
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  UnderlyingGeneralizedMorphism( phi )  attribute
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  Returns:  a morphism
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  For a morphism phi in the Serre quotient category A/C this attribute returns
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  the  corresponding generalized morphism in the generalized morphism category
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  the quotient is modelled upon.
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  7.1-17 CanonicalProjection
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  CanonicalProjection( A/C )  attribute
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  Returns:  a functor
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  Given  a  Serre  quotient category A/C, this operation returns the canonical
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  projection functor  A \rightarrow A/C .
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  7.2 Serre quotients by cospans
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  7.2-1 SerreQuotientCategoryByCospans
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  SerreQuotientCategoryByCospans( A, func[, name] )  operation
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  Returns:  a CAP category
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  Creates  a  Serre  quotient  category  S  with  name  name out of an Abelian
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  category A. The Serre quotient category will be modeled upon the generalized
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  morphisms  by  cospans category of A If name is not given, a generic name is
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  constructed out of the name of A. The argument func must be a unary function
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  on  the  objects  of  A  deciding  the membership in the thick subcategory C
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  mentioned above.
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  7.2-2 AsSerreQuotientCategoryByCospansObject
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  AsSerreQuotientCategoryByCospansObject( A/C, M )  operation
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  Returns:  an object
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  Given a Serre quotient category A/C modeled by cospans and an object M in A,
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  this  constructor  returns  the  corresponding  object in the Serre quotient
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  category.
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  7.2-3 SerreQuotientCategoryByCospansMorphism
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  SerreQuotientCategoryByCospansMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category A/C modeled by cospans and a generalized
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  morphism  phi in the generalized morphism category A/C is modeled upon, this
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  constructor  returns  the  corresponding  morphism  in  the  Serre  quotient
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  category.
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  7.2-4 SerreQuotientCategoryByCospansMorphism
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  SerreQuotientCategoryByCospansMorphism( A/C, iota, phi, pi )  operation
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  Returns:  a morphism
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  Given  a  Serre quotient category A/C modeled by cospans and three morphisms
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  \iota:  M'  \rightarrow M, \phi: M' \rightarrow N' and \pi: N \rightarrow N'
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  this operation contructs a morphism in the Serre quotient category.
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  7.2-5 SerreQuotientCategoryByCospansMorphismWithSourceAid
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  SerreQuotientCategoryByCospansMorphismWithSourceAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category A/C modeled by cospans and two morphisms
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  \alpha: M \rightarrow X and \beta: X \rightarrow N this operation constructs
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  the corresponding morphism in the Serre quotient category.
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  7.2-6 SerreQuotientCategoryByCospansMorphism
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  SerreQuotientCategoryByCospansMorphism( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category A/C modeled by cospans and two morphisms
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  \alpha: X \rightarrow M and \beta: X \rightarrow N this operation constructs
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  the corresponding morphism in the Serre quotient category.
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  7.2-7 AsSerreQuotientCategoryByCospansMorphism
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  AsSerreQuotientCategoryByCospansMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given a Serre quotient category A/C modeled by cospans and a morphism phi in
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  A, this constructor returns the corresponding morphism in the Serre quotient
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  category.
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  7.3 Serre Quotients by Spans
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  7.3-1 SerreQuotientCategoryBySpans
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  SerreQuotientCategoryBySpans( A, func[, name] )  operation
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  Returns:  a CAP category
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  Creates  a  Serre  quotient  category  S  with  name  name out of an Abelian
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  category A. The Serre quotient category will be modeled upon the generalized
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  morphisms  by  spans  category  of A If name is not given, a generic name is
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  constructed out of the name of A. The argument func must be a unary function
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  on  the  objects  of  A  deciding  the membership in the thick subcategory C
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  mentioned above.
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  7.3-2 AsSerreQuotientCategoryBySpansObject
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  AsSerreQuotientCategoryBySpansObject( A/C, M )  operation
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  Returns:  an object
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  Given  a  Serre quotient category A/C modeled by spans and an object M in A,
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  this  constructor  returns  the  corresponding  object in the Serre quotient
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  category.
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  7.3-3 SerreQuotientCategoryBySpansMorphism
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  SerreQuotientCategoryBySpansMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled by spans and a generalized
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  morphism  phi in the generalized morphism category A/C is modeled upon, this
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  constructor  returns  the  corresponding  morphism  in  the  Serre  quotient
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  category.
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  7.3-4 SerreQuotientCategoryBySpansMorphism
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  SerreQuotientCategoryBySpansMorphism( A/C, iota, phi, pi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category A/C modeled by spans and three morphisms
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  \iota:  M'  \rightarrow M, \phi: M' \rightarrow N' and \pi: N \rightarrow N'
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  this operation contructs a morphism in the Serre quotient category.
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  7.3-5 SerreQuotientCategoryBySpansMorphism
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  SerreQuotientCategoryBySpansMorphism( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled by spans and two morphisms
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  \alpha: M \rightarrow X and \beta: X \rightarrow N this operation constructs
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  the corresponding morphism in the Serre quotient category.
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  7.3-6 SerreQuotientCategoryBySpansMorphismWithRangeAid
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  SerreQuotientCategoryBySpansMorphismWithRangeAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled by spans and two morphisms
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  \alpha: X \rightarrow M and \beta: X \rightarrow N this operation constructs
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  the corresponding morphism in the Serre quotient category.
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  7.3-7 AsSerreQuotientCategoryBySpansMorphism
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  AsSerreQuotientCategoryBySpansMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre quotient category A/C modeled by spans and a morphism phi in
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  A, this constructor returns the corresponding morphism in the Serre quotient
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  category.
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  7.4 Serre Quotients modeled by three arrows
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  7.4-1 SerreQuotientCategoryByThreeArrows
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  SerreQuotientCategoryByThreeArrows( A, func[, name] )  operation
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  Returns:  a CAP category
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  Creates  a  Serre  quotient  category  S  with  name  name out of an Abelian
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  category A. The Serre quotient category will be modeled upon the generalized
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  morphisms by three arrows category of A If name is not given, a generic name
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  is  constructed  out  of  the  name  of A. The argument func must be a unary
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  function  on  the  objects  of  A  deciding  the  membership  in  the  thick
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  subcategory C mentioned above.
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  7.4-2 AsSerreQuotientCategoryByThreeArrowsObject
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  AsSerreQuotientCategoryByThreeArrowsObject( A/C, M )  operation
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  Returns:  an object
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  Given  a Serre quotient category A/C modeled by three arrows and an object M
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  in  A,  this  constructor  returns  the  corresponding  object  in the Serre
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  quotient category.
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  7.4-3 SerreQuotientCategoryByThreeArrowsMorphism
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  SerreQuotientCategoryByThreeArrowsMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled  by  three  arrows  and  a
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  generalized morphism phi in the generalized morphism category A/C is modeled
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  upon,  this  constructor  returns  the  corresponding  morphism in the Serre
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  quotient category.
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  7.4-4 SerreQuotientCategoryByThreeArrowsMorphism
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  SerreQuotientCategoryByThreeArrowsMorphism( A/C, iota, phi, pi )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled  by three arrows and three
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  morphisms  \iota:  M'  \rightarrow  M,  \phi:  M'  \rightarrow N' and \pi: N
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  \rightarrow  N'  this  operation  contructs a morphism in the Serre quotient
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  category.
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  7.4-5 SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid
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  SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled  by  three  arrows and two
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  morphisms  \alpha: M \rightarrow X and \beta: X \rightarrow N this operation
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  constructs the corresponding morphism in the Serre quotient category.
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  7.4-6 SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid
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  SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid( A/C, alpha, beta )  operation
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  Returns:  a morphism
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  Given  a  Serre  quotient  category  A/C  modeled  by  three  arrows and two
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  morphisms  \alpha: X \rightarrow M and \beta: X \rightarrow N this operation
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  constructs the corresponding morphism in the Serre quotient category.
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  7.4-7 AsSerreQuotientCategoryByThreeArrowsMorphism
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  AsSerreQuotientCategoryByThreeArrowsMorphism( A/C, phi )  operation
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  Returns:  a morphism
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  Given  a  Serre quotient category A/C modeled by three arrows and a morphism
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  phi  in  A, this constructor returns the corresponding morphism in the Serre
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  quotient category.
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