GAP (GeneralizedMorphismsForCAP) - Chapter 1: Generalized Morphism Category

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1 Generalized Morphism Category
1.1 GAP Categories
1.1-1 IsGeneralizedMorphismCategoryObject
1.1-2 IsGeneralizedMorphism
1.2 Attributes
1.2-1 UnderlyingHonestObject
1.2-2 DomainEmbedding
1.2-3 GeneralizedImageEmbedding
1.2-4 DefectEmbedding
1.2-5 GeneralizedKernelEmbedding
1.2-6 CodomainProjection
1.2-7 GeneralizedCokernelProjection
1.2-8 CodefectProjection
1.2-9 GeneralizedCoimageProjection
1.2-10 AssociatedMorphism
1.2-11 DomainAssociatedMorphismCodomainTriple
1.2-12 HonestRepresentative
1.2-13 GeneralizedInverse
1.2-14 IdempotentDefinedBySubobject
1.2-15 IdempotentDefinedByFactorobject
1.2-16 UnderlyingHonestCategory
1.3 Operations
1.3-1 GeneralizedMorphismFromFactorToSubobject
1.3-2 CommonRestriction
1.3-3 ConcatenationProduct
1.4 Properties
1.4-1 IsHonest
1.4-2 HasFullDomain
1.4-3 HasFullCodomain
1.4-4 IsSingleValued
1.4-5 IsTotal
1.5 Convenience methods
1.5-1 GeneralizedMorphismCategory
1.5-2 GeneralizedMorphismObject
1.5-3 AsGeneralizedMorphism
1.5-4 GeneralizedMorphism
1.5-5 GeneralizedMorphism
1.5-6 GeneralizedMorphismWithRangeAid
1.5-7 GeneralizedMorphismWithSourceAid

### 1 Generalized Morphism Category

Let \mathbf{A} be an abelian category. We denote its generalized morphism category by \mathbf{G(A)}.

#### 1.1 GAP Categories

##### 1.1-1 IsGeneralizedMorphismCategoryObject

`‣ IsGeneralizedMorphismCategoryObject`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the generalized morphism category.

##### 1.1-2 IsGeneralizedMorphism

`‣ IsGeneralizedMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the generalized morphism category.

#### 1.2 Attributes

##### 1.2-1 UnderlyingHonestObject

`‣ UnderlyingHonestObject`( a ) | ( attribute )

Returns: an object in \mathbf{A}

The argument is an object a in the generalized morphism category. The output is its underlying honest object

##### 1.2-2 DomainEmbedding

`‣ DomainEmbedding`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( d, a )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its domain d \hookrightarrow a \in \mathbf{A}.

##### 1.2-3 GeneralizedImageEmbedding

`‣ GeneralizedImageEmbedding`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( i, b )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its generalized image i \hookrightarrow b \in \mathbf{A}.

##### 1.2-4 DefectEmbedding

`‣ DefectEmbedding`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( d, b )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its defect d \hookrightarrow b \in \mathbf{A}.

##### 1.2-5 GeneralizedKernelEmbedding

`‣ GeneralizedKernelEmbedding`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( k, a )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its generalized kernel k \hookrightarrow a \in \mathbf{A}.

##### 1.2-6 CodomainProjection

`‣ CodomainProjection`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( b, c )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its codomain b \twoheadrightarrow c \in \mathbf{A}.

##### 1.2-7 GeneralizedCokernelProjection

`‣ GeneralizedCokernelProjection`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( b, c )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its generalized cokernel b \twoheadrightarrow c \in \mathbf{A}.

##### 1.2-8 CodefectProjection

`‣ CodefectProjection`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( a, c )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its codefect a \twoheadrightarrow c \in \mathbf{A}.

##### 1.2-9 GeneralizedCoimageProjection

`‣ GeneralizedCoimageProjection`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( a, c )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its generalized coimage a \twoheadrightarrow c \in \mathbf{A}.

##### 1.2-10 AssociatedMorphism

`‣ AssociatedMorphism`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( d, c )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is its associated morphism d \rightarrow c \in \mathbf{A}.

##### 1.2-11 DomainAssociatedMorphismCodomainTriple

`‣ DomainAssociatedMorphismCodomainTriple`( alpha ) | ( attribute )

Returns: a triple of morphisms in \mathbf{A}

The argument is a generalized morphism \alpha: a \rightarrow b. The output is a triple ( d \hookrightarrow a, d \rightarrow c, b \twoheadrightarrow c ) consisting of its domain, associated morphism, and codomain.

##### 1.2-12 HonestRepresentative

`‣ HonestRepresentative`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}( a, b )

The argument is a generalized morphism \alpha: a \rightarrow b. The output is the honest representative in \mathbf{A} of \alpha, if it exists, otherwise an error is thrown.

##### 1.2-13 GeneralizedInverse

`‣ GeneralizedInverse`( alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a)

The argument is a morphism \alpha: a \rightarrow b \in \mathbf{A}. The output is its generalized inverse b \rightarrow a.

##### 1.2-14 IdempotentDefinedBySubobject

`‣ IdempotentDefinedBySubobject`( alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b)

The argument is a subobject \alpha: a \hookrightarrow b \in \mathbf{A}. The output is the idempotent b \rightarrow b \in \mathbf{G(A)} defined by \alpha.

##### 1.2-15 IdempotentDefinedByFactorobject

`‣ IdempotentDefinedByFactorobject`( alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b)

The argument is a factorobject \alpha: b \twoheadrightarrow a \in \mathbf{A}. The output is the idempotent b \rightarrow b \in \mathbf{G(A)} defined by \alpha.

##### 1.2-16 UnderlyingHonestCategory

`‣ UnderlyingHonestCategory`( C ) | ( attribute )

Returns: a category

The argument is a generalized morphism category C = \mathbf{G(A)}. The output is \mathbf{A}.

#### 1.3 Operations

##### 1.3-1 GeneralizedMorphismFromFactorToSubobject

`‣ GeneralizedMorphismFromFactorToSubobject`( beta, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(c,a)

The arguments are a a factorobject \beta: b \twoheadrightarrow c, and a subobject \alpha: a \hookrightarrow b. The output is the generalized morphism from the factorobject to the subobject.

##### 1.3-2 CommonRestriction

`‣ CommonRestriction`( L ) | ( operation )

Returns: a list of generalized morphisms

The argument is a list L of generalized morphisms by three arrows having the same source. The output is a list of generalized morphisms by three arrows which is the comman restriction of L.

##### 1.3-3 ConcatenationProduct

`‣ ConcatenationProduct`( L ) | ( operation )

Returns: a generalized moprhism

The argument is a list L = ( \alpha_1, \dots, \alpha_n ) of generalized morphisms (with same data structures). The output is their concatenation product, i.e., a generalized morphism \alpha with \mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha_i ) ), and \mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha_i ) ), and with morphisms in the representation of \alpha given as the direct sums of the corresponding morphisms of the \alpha_i.

#### 1.4 Properties

##### 1.4-1 IsHonest

`‣ IsHonest`( alpha ) | ( property )

Returns: a boolean

The argument is a generalized morphism \alpha. The output is `true` if \alpha admits an honest representative, otherwise `false`.

##### 1.4-2 HasFullDomain

`‣ HasFullDomain`( alpha ) | ( property )

Returns: a boolean

The argument is a generalized morphism \alpha. The output is `true` if the domain of \alpha is an isomorphism, otherwise `false`.

##### 1.4-3 HasFullCodomain

`‣ HasFullCodomain`( alpha ) | ( property )

Returns: a boolean

The argument is a generalized morphism \alpha. The output is `true` if the codomain of \alpha is an isomorphism, otherwise `false`.

##### 1.4-4 IsSingleValued

`‣ IsSingleValued`( alpha ) | ( property )

Returns: a boolean

The argument is a generalized morphism \alpha. The output is `true` if the codomain of \alpha is an isomorphism, otherwise `false`.

##### 1.4-5 IsTotal

`‣ IsTotal`( alpha ) | ( property )

Returns: a boolean

The argument is a generalized morphism \alpha. The output is `true` if the domain of \alpha is an isomorphism, otherwise `false`.

#### 1.5 Convenience methods

This section contains operations which, depending on the current generalized morphism standard of the system and the category, might point to other Operations. Please use them only as convenience and never in serious code.

##### 1.5-1 GeneralizedMorphismCategory

`‣ GeneralizedMorphismCategory`( C ) | ( operation )

Returns: a category

Creates a new category of generalized morphisms. Might point to GeneralizedMorphismCategoryByThreeArrows, GeneralizedMorphismCategoryByCospans, or GeneralizedMorphismCategoryBySpans

##### 1.5-2 GeneralizedMorphismObject

`‣ GeneralizedMorphismObject`( A ) | ( operation )

Returns: an object in the generalized morphism category

Creates an object in the current generalized morphism category, depending on the standard

##### 1.5-3 AsGeneralizedMorphism

`‣ AsGeneralizedMorphism`( phi ) | ( operation )

Returns: a generalized morphism

Returns the corresponding morphism to phi in the current generalized morphism category.

##### 1.5-4 GeneralizedMorphism

`‣ GeneralizedMorphism`( phi, psi ) | ( operation )

Returns: a generalized morphism

Returns the corresponding morphism to phi and psi in the current generalized morphism category.

##### 1.5-5 GeneralizedMorphism

`‣ GeneralizedMorphism`( iota, phi, pi ) | ( operation )

Returns: a generalized morphism

Returns the corresponding morphism to iota, phi and psi in the current generalized morphism category.

##### 1.5-6 GeneralizedMorphismWithRangeAid

`‣ GeneralizedMorphismWithRangeAid`( arg1, arg2 ) | ( operation )

Returns a generalized morphism with range aid by three arrows or by span, or a generalized morphism by cospan, depending on the standard.

##### 1.5-7 GeneralizedMorphismWithSourceAid

`‣ GeneralizedMorphismWithSourceAid`( arg1, arg2 ) | ( operation )

Returns a generalized morphism with source aid by three arrows or by cospan, or a generalized morphism by span, depending on the standard.

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