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### 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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