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7 Serre Quotients
 7.1 General operations
 7.2 Serre quotients by cospans
 7.3 Serre Quotients by Spans
 7.4 Serre Quotients modeled by three arrows

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.

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