GAP (GeneralizedMorphismsForCAP) - Chapter 4: Generalized Morphism Category by Three Arrows

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4 Generalized Morphism Category by Three Arrows
4.1 GAP Categories
4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject
4.1-2 IsGeneralizedMorphismByThreeArrows
4.2 Properties
4.2-1 HasIdentitiesAsReversedArrows
4.2-2 HasIdentityAsSourceAid
4.2-3 HasIdentityAsRangeAid
4.3 Attributes
4.3-1 UnderlyingHonestObject
4.3-2 SourceAid
4.3-3 RangeAid
4.3-4 Arrow
4.3-5 PseudoInverse
4.3-6 GeneralizedInverseByThreeArrows
4.3-7 IdempotentDefinedBySubobjectByThreeArrows
4.3-8 IdempotentDefinedByFactorobjectByThreeArrows
4.4 Operations
4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows
4.4-2 CommonCoastriction
4.5 Constructors
4.5-1 GeneralizedMorphismByThreeArrows
4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid
4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid
4.5-4 AsGeneralizedMorphismByThreeArrows
4.5-5 GeneralizedMorphismCategoryByThreeArrows
4.5-6 GeneralizedMorphismByThreeArrowsObject

### 4 Generalized Morphism Category by Three Arrows

#### 4.1 GAP Categories

##### 4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject

`‣ IsGeneralizedMorphismCategoryByThreeArrowsObject`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the generalized morphism category by three arrows.

##### 4.1-2 IsGeneralizedMorphismByThreeArrows

`‣ IsGeneralizedMorphismByThreeArrows`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the generalized morphism category by three arrows.

#### 4.2 Properties

##### 4.2-1 HasIdentitiesAsReversedArrows

`‣ HasIdentitiesAsReversedArrows`( alpha ) | ( property )

Returns: `true` or `false`

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is `true` if \(a \leftarrow b\) and \(c \leftarrow d\) are congruent to identity morphisms, `false` otherwise.

##### 4.2-2 HasIdentityAsSourceAid

`‣ HasIdentityAsSourceAid`( alpha ) | ( property )

Returns: `true` or `false`

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is `true` if \(a \leftarrow b\) is congruent to an identity morphism, `false` otherwise.

##### 4.2-3 HasIdentityAsRangeAid

`‣ HasIdentityAsRangeAid`( alpha ) | ( property )

Returns: `true` or `false`

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is `true` if \(c \leftarrow d\) is congruent to an identity morphism, `false` otherwise.

#### 4.3 Attributes

##### 4.3-1 UnderlyingHonestObject

`‣ UnderlyingHonestObject`( a ) | ( attribute )

Returns: an object in \(\mathbf{A}\)

The argument is an object \(a\) in the generalized morphism category by three arrows. The output is its underlying honest object.

##### 4.3-2 SourceAid

`‣ SourceAid`( alpha ) | ( attribute )

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

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its source aid \(a \leftarrow b\).

##### 4.3-3 RangeAid

`‣ RangeAid`( alpha ) | ( attribute )

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

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its range aid \(c \leftarrow d\).

##### 4.3-4 Arrow

`‣ Arrow`( alpha ) | ( attribute )

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

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its range aid \(b \rightarrow c\).

##### 4.3-5 PseudoInverse

`‣ PseudoInverse`( alpha ) | ( attribute )

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

The argument is a generalized morphism \(\alpha: a \rightarrow b\) by three arrows. The output is its pseudo inverse \(b \rightarrow a\).

##### 4.3-6 GeneralizedInverseByThreeArrows

`‣ GeneralizedInverseByThreeArrows`( alpha ) | ( attribute )

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 \(b \rightarrow a\) by three arrows.

##### 4.3-7 IdempotentDefinedBySubobjectByThreeArrows

`‣ IdempotentDefinedBySubobjectByThreeArrows`( alpha ) | ( attribute )

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 \(b \rightarrow b \in \mathbf{G(A)}\) by three arrows defined by \(\alpha\).

##### 4.3-8 IdempotentDefinedByFactorobjectByThreeArrows

`‣ IdempotentDefinedByFactorobjectByThreeArrows`( alpha ) | ( attribute )

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 \(b \rightarrow b \in \mathbf{G(A)}\) by three arrows defined by \(\alpha\).

#### 4.4 Operations

##### 4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows

`‣ GeneralizedMorphismFromFactorToSubobjectByThreeArrows`( 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 by three arrows from the factorobject to the subobject.

##### 4.4-2 CommonCoastriction

`‣ CommonCoastriction`( L ) | ( operation )

Returns: a list of generalized morphisms

The argument is a list \(L\) of generalized morphisms by three arrows having the same range. The output is a list of generalized morphisms by three arrows which is the comman coastriction of \(L\).

#### 4.5 Constructors

##### 4.5-1 GeneralizedMorphismByThreeArrows

`‣ GeneralizedMorphismByThreeArrows`( alpha, beta, gamma ) | ( operation )

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

The arguments are morphisms \(\alpha: a \leftarrow b\), \(\beta: b \rightarrow c\), and \(\gamma: c \leftarrow d\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows with source aid \(\alpha\), arrow \(\beta\), and range aid \(\gamma\).

##### 4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid

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

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

The arguments are morphisms \(\alpha: a \leftarrow b\), and \(\beta: b \rightarrow c\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows defined by the composition the given two arrows regarded as generalized morphisms.

##### 4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid

`‣ GeneralizedMorphismByThreeArrowsWithRangeAid`( beta, gamma ) | ( operation )

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

The arguments are morphisms \(\beta: b \rightarrow c\), and \(\gamma: c \leftarrow d\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows defined by the composition the given two arrows regarded as generalized morphisms.

##### 4.5-4 AsGeneralizedMorphismByThreeArrows

`‣ AsGeneralizedMorphismByThreeArrows`( alpha ) | ( attribute )

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

The argument is a morphism \(\alpha: a \rightarrow b\) in \(\mathbf{A}\). The output is the honest generalized morphism by three arrows defined by \(\alpha\).

##### 4.5-5 GeneralizedMorphismCategoryByThreeArrows

`‣ GeneralizedMorphismCategoryByThreeArrows`( A ) | ( attribute )

Returns: a category

The argument is an abelian category \(\mathbf{A}\). The output is its generalized morphism category \(\mathbf{G(A)}\) by three arrows.

##### 4.5-6 GeneralizedMorphismByThreeArrowsObject

`‣ GeneralizedMorphismByThreeArrowsObject`( a ) | ( attribute )

Returns: an object in \(\mathbf{G(A)}\)

The argument is an object \(a\) in an abelian category \(\mathbf{A}\). The output is the object in the generalized morphism category by three arrows whose underlying honest object is \(a\).

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