chap7.txt

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