GAP (GeneralizedMorphismsForCAP) - Chapter 7: Serre Quotients

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7 Serre Quotients
7.1 General operations
7.1-1 IsSerreQuotientCategoryObject
7.1-2 IsSerreQuotientCategoryMorphism
7.1-3 SerreQuotientCategory
7.1-4 AsSerreQuotientCategoryObject
7.1-5 SerreQuotientCategoryMorphism
7.1-6 SerreQuotientCategoryMorphism
7.1-7 SerreQuotientCategoryMorphism
7.1-8 SerreQuotientCategoryMorphismWithSourceAid
7.1-9 SerreQuotientCategoryMorphismWithRangeAid
7.1-10 AsSerreQuotientCategoryMorphism
7.1-11 SubcategoryMembershipTestFunctionForSerreQuotient
7.1-12 UnderlyingHonestCategory
7.1-13 UnderlyingGeneralizedMorphismCategory
7.1-14 UnderlyingGeneralizedObject
7.1-15 UnderlyingHonestObject
7.1-16 UnderlyingGeneralizedMorphism
7.1-17 CanonicalProjection
7.2 Serre quotients by cospans
7.2-1 SerreQuotientCategoryByCospans
7.2-2 AsSerreQuotientCategoryByCospansObject
7.2-3 SerreQuotientCategoryByCospansMorphism
7.2-4 SerreQuotientCategoryByCospansMorphism
7.2-5 SerreQuotientCategoryByCospansMorphismWithSourceAid
7.2-6 SerreQuotientCategoryByCospansMorphism
7.2-7 AsSerreQuotientCategoryByCospansMorphism
7.3 Serre Quotients by Spans
7.3-1 SerreQuotientCategoryBySpans
7.3-2 AsSerreQuotientCategoryBySpansObject
7.3-3 SerreQuotientCategoryBySpansMorphism
7.3-4 SerreQuotientCategoryBySpansMorphism
7.3-5 SerreQuotientCategoryBySpansMorphism
7.3-6 SerreQuotientCategoryBySpansMorphismWithRangeAid
7.3-7 AsSerreQuotientCategoryBySpansMorphism
7.4 Serre Quotients modeled by three arrows
7.4-1 SerreQuotientCategoryByThreeArrows
7.4-2 AsSerreQuotientCategoryByThreeArrowsObject
7.4-3 SerreQuotientCategoryByThreeArrowsMorphism
7.4-4 SerreQuotientCategoryByThreeArrowsMorphism
7.4-5 SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid
7.4-6 SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid
7.4-7 AsSerreQuotientCategoryByThreeArrowsMorphism

### 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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