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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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