‣ 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 \(A \to B\), where \(A := \) GeneralizedMorphismCategoryByCospans( AsCapCategory( Source( F ) ) ) and \(B := \) GeneralizedMorphismCategoryByCospans( AsCapCategory( Range( F ) ) ).
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