‣ IsGeneralizedMorphismCategoryByCospans( object ) | ( filter ) |
Returns: true or false
The GAP category of the category of generalized morphisms by cospans.
‣ IsGeneralizedMorphismCategoryByCospansObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the generalized morphism category by cospans.
‣ IsGeneralizedMorphismByCospan( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the generalized morphism category by cospans.
‣ HasIdentityAsReversedArrow( alpha ) | ( property ) |
Returns: true or false
The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is true if b \leftarrow c 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 cospans. The output is its underlying honest object.
‣ Arrow( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}(a,c)
The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is its arrow a \rightarrow b.
‣ ReversedArrow( alpha ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}_{\mathbf{A}}(c,b)
The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is its reversed arrow b \leftarrow c.
‣ NormalizedCospanTuple( alpha ) | ( attribute ) |
Returns: a pair of morphisms in \mathbf{A}.
The argument is a generalized morphism \alpha: a \rightarrow b by a cospan. The output is its normalized cospan pair (a \rightarrow d, d \leftarrow 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 cospan. The output is its pseudo inverse b \rightarrow a.
‣ GeneralizedInverseByCospan( 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 cospan.
‣ IdempotentDefinedBySubobjectByCospan( 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 cospan defined by \alpha.
‣ IdempotentDefinedByFactorobjectByCospan( 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 cospan defined by \alpha.
‣ NormalizedCospan( 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 cospan. The output is its normalization by cospan.
‣ GeneralizedMorphismFromFactorToSubobjectByCospan( 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 cospan from the factorobject to the subobject.
‣ GeneralizedMorphismByCospan( alpha, beta ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c)
The arguments are morphisms \alpha: a \rightarrow b and \beta: c \rightarrow b in \mathbf{A}. The output is a generalized morphism by cospan with arrow \alpha and reversed arrow \beta.
‣ GeneralizedMorphismByCospan( 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 cospan defined by the composition of the given three arrows regarded as generalized morphisms.
‣ GeneralizedMorphismByCospanWithSourceAid( 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 cospan defined by the composition of the given two arrows regarded as generalized morphisms.
‣ AsGeneralizedMorphismByCospan( 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 cospan defined by \alpha.
‣ GeneralizedMorphismCategoryByCospans( A ) | ( attribute ) |
Returns: a category
The argument is an abelian category \mathbf{A}. The output is its generalized morphism category \mathbf{G(A)} by cospans.
‣ GeneralizedMorphismByCospansObject( 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 cospans whose underlying honest object is a.
‣ AsGeneralizedMorphismByCospan( F, name ) | ( operation ) |
Lift the exact functor F to a functor A -> B, where A := GeneralizedMorphismCategoryByCospans( AsCapCategory( Source( F ) ) ) and B := GeneralizedMorphismCategoryByCospans( AsCapCategory( Range( F ) ) ).
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