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4 Generalized Morphism Category by Three Arrows
 4.1 GAP Categories
 4.2 Properties
 4.3 Attributes
 4.4 Operations
 4.5 Constructors

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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