‣ 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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