‣ IsGeneralizedMorphismCategoryByThreeArrows( object ) | ( filter ) |
Returns: true or false
The GAP category of the category of generalized morphisms by three arrows.
‣ IsGeneralizedMorphismCategoryByThreeArrowsObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the generalized morphism category by three arrows.
‣ IsGeneralizedMorphismByThreeArrows( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the generalized morphism category by three arrows.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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 of the given two arrows regarded as generalized morphisms.
‣ 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 of the given two arrows regarded as generalized morphisms.
‣ 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.
‣ 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.
‣ 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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