‣ IsGeneralizedMorphismCategoryBySpans( object ) | ( filter ) |
Returns: true or false
The GAP category of the category of generalized morphisms by spans.
‣ IsGeneralizedMorphismCategoryBySpansObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the generalized morphism category by spans.
‣ IsGeneralizedMorphismBySpan( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the generalized morphism category by spans.
‣ HasIdentityAsReversedArrow( alpha ) | ( property ) |
Returns: true or false
The argument is a generalized morphism \(\alpha\) by a span \(a \leftarrow b \rightarrow c\). The output is true if \(a \leftarrow b\) 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 spans. The output is its underlying honest object.
‣ Arrow( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}(b,c)\)
The argument is a generalized morphism \(\alpha\) by a span \(a \leftarrow b \rightarrow c\). The output is its arrow \(b \rightarrow c\).
‣ ReversedArrow( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}(b,a)\)
The argument is a generalized morphism \(\alpha\) by a span \(a \leftarrow b \rightarrow c\). The output is its reversed arrow \(a \leftarrow b\).
‣ NormalizedSpanTuple( alpha ) | ( attribute ) |
Returns: a pair of morphisms in \(\mathbf{A}\).
The argument is a generalized morphism \(\alpha: a \rightarrow b\) by a span. The output is its normalized span pair \((a \leftarrow d, d \rightarrow b)\).
‣ 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 a span. The output is its pseudo inverse \(b \rightarrow a\).
‣ GeneralizedInverseBySpan( 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 span.
‣ IdempotentDefinedBySubobjectBySpan( 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 span defined by \(\alpha\).
‣ IdempotentDefinedByFactorobjectBySpan( 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 span defined by \(\alpha\).
‣ NormalizedSpan( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,b)\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\) by a span. The output is its normalization by span.
‣ GeneralizedMorphismFromFactorToSubobjectBySpan( 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 span from the factorobject to the subobject.
‣ GeneralizedMorphismBySpan( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,b)\)
The arguments are morphisms \(\alpha: a \leftarrow c\) and \(\beta: c \rightarrow b\) in \(\mathbf{A}\). The output is a generalized morphism by span with arrow \(\beta\) and reversed arrow \(\alpha\).
‣ GeneralizedMorphismBySpan( 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 span defined by the composition of the given three arrows regarded as generalized morphisms.
‣ GeneralizedMorphismBySpanWithRangeAid( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,c)\)
The arguments are morphisms \(\alpha: a \rightarrow b\), and \(\beta: b \leftarrow c\) in \(\mathbf{A}\). The output is a generalized morphism by span defined by the composition of the given two arrows regarded as generalized morphisms.
‣ AsGeneralizedMorphismBySpan( 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 span defined by \(\alpha\).
‣ GeneralizedMorphismCategoryBySpans( A ) | ( attribute ) |
Returns: a category
The argument is an abelian category \(\mathbf{A}\). The output is its generalized morphism category \(\mathbf{G(A)}\) by spans.
‣ GeneralizedMorphismBySpansObject( 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 spans whose underlying honest object is \(a\).
generated by GAPDoc2HTML