‣ 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.
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