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### 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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