Let \(\mathbf{A}\) be an abelian category. We denote its generalized morphism category by \(\mathbf{G(A)}\).
‣ IsGeneralizedMorphismCategory( object ) | ( filter ) |
Returns: true or false
The GAP category of the category of generalized morphisms.
‣ IsGeneralizedMorphismCategoryObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the generalized morphism category.
‣ IsGeneralizedMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the generalized morphism category.
‣ UnderlyingHonestObject( a ) | ( attribute ) |
Returns: an object in \(\mathbf{A}\)
The argument is an object \(a\) in the generalized morphism category. The output is its underlying honest object
‣ DomainEmbedding( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( d, a )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its domain \(d \hookrightarrow a \in \mathbf{A}\).
‣ GeneralizedImageEmbedding( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( i, b )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its generalized image \(i \hookrightarrow b \in \mathbf{A}\).
‣ DefectEmbedding( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( d, b )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its defect \(d \hookrightarrow b \in \mathbf{A}\).
‣ GeneralizedKernelEmbedding( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( k, a )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its generalized kernel \(k \hookrightarrow a \in \mathbf{A}\).
‣ CodomainProjection( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( b, c )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its codomain \(b \twoheadrightarrow c \in \mathbf{A}\).
‣ GeneralizedCokernelProjection( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( b, c )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its generalized cokernel \(b \twoheadrightarrow c \in \mathbf{A}\).
‣ CodefectProjection( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( a, c )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its codefect \(a \twoheadrightarrow c \in \mathbf{A}\).
‣ GeneralizedCoimageProjection( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( a, c )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its generalized coimage \(a \twoheadrightarrow c \in \mathbf{A}\).
‣ AssociatedMorphism( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( d, c )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is its associated morphism \(d \rightarrow c \in \mathbf{A}\).
‣ DomainAssociatedMorphismCodomainTriple( alpha ) | ( attribute ) |
Returns: a triple of morphisms in \(\mathbf{A}\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is a triple \(( d \hookrightarrow a, d \rightarrow c, b \twoheadrightarrow c )\) consisting of its domain, associated morphism, and codomain.
‣ HonestRepresentative( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}( a, b )\)
The argument is a generalized morphism \(\alpha: a \rightarrow b\). The output is the honest representative in \(\mathbf{A}\) of \(\alpha\), if it exists, otherwise an error is thrown.
‣ GeneralizedInverse( alpha ) | ( operation ) |
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 \(b \rightarrow a\).
‣ IdempotentDefinedBySubobject( alpha ) | ( operation ) |
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 \(b \rightarrow b \in \mathbf{G(A)}\) defined by \(\alpha\).
‣ IdempotentDefinedByFactorobject( alpha ) | ( operation ) |
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 \(b \rightarrow b \in \mathbf{G(A)}\) defined by \(\alpha\).
‣ UnderlyingHonestCategory( C ) | ( attribute ) |
Returns: a category
The argument is a generalized morphism category \(C = \mathbf{G(A)}\). The output is \(\mathbf{A}\).
‣ GeneralizedMorphismFromFactorToSubobject( 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 from the factorobject to the subobject.
‣ CommonRestriction( L ) | ( operation ) |
Returns: a list of generalized morphisms
The argument is a list \(L\) of generalized morphisms by three arrows having the same source. The output is a list of generalized morphisms by three arrows which is the comman restriction of \(L\).
‣ ConcatenationProduct( L ) | ( operation ) |
Returns: a generalized moprhism
The argument is a list \(L = ( \alpha_1, \dots, \alpha_n )\) of generalized morphisms (with same data structures). The output is their concatenation product, i.e., a generalized morphism \(\alpha\) with \(\mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha_i ) )\), and \(\mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha_i ) )\), and with morphisms in the representation of \(\alpha\) given as the direct sums of the corresponding morphisms of the \(\alpha_i\).
‣ IsHonest( alpha ) | ( property ) |
Returns: a boolean
The argument is a generalized morphism \(\alpha\). The output is true if \(\alpha\) admits an honest representative, otherwise false.
‣ HasFullDomain( alpha ) | ( property ) |
Returns: a boolean
The argument is a generalized morphism \(\alpha\). The output is true if the domain of \(\alpha\) is an isomorphism, otherwise false.
‣ HasFullCodomain( alpha ) | ( property ) |
Returns: a boolean
The argument is a generalized morphism \(\alpha\). The output is true if the codomain of \(\alpha\) is an isomorphism, otherwise false.
‣ IsSingleValued( alpha ) | ( property ) |
Returns: a boolean
The argument is a generalized morphism \(\alpha\). The output is true if the codomain of \(\alpha\) is an isomorphism, otherwise false.
‣ IsTotal( alpha ) | ( property ) |
Returns: a boolean
The argument is a generalized morphism \(\alpha\). The output is true if the domain of \(\alpha\) is an isomorphism, otherwise false.
This section contains operations which, depending on the current generalized morphism standard of the system and the category, might point to other Operations. Please use them only as convenience and never in serious code.
‣ GeneralizedMorphismCategory( C ) | ( operation ) |
Returns: a category
Creates a new category of generalized morphisms. Might point to GeneralizedMorphismCategoryByThreeArrows, GeneralizedMorphismCategoryByCospans, or GeneralizedMorphismCategoryBySpans
‣ GeneralizedMorphismObject( A ) | ( operation ) |
Returns: an object in the generalized morphism category
Creates an object in the current generalized morphism category, depending on the standard
‣ AsGeneralizedMorphism( phi ) | ( operation ) |
Returns: a generalized morphism
Returns the corresponding morphism to phi in the current generalized morphism category.
‣ GeneralizedMorphism( phi, psi ) | ( operation ) |
Returns: a generalized morphism
Returns the corresponding morphism to phi and psi in the current generalized morphism category.
‣ GeneralizedMorphism( iota, phi, pi ) | ( operation ) |
Returns: a generalized morphism
Returns the corresponding morphism to iota, phi and psi in the current generalized morphism category.
‣ GeneralizedMorphismWithRangeAid( arg1, arg2 ) | ( operation ) |
Returns a generalized morphism with range aid by three arrows or by span, or a generalized morphism by cospan, depending on the standard.
‣ GeneralizedMorphismWithSourceAid( arg1, arg2 ) | ( operation ) |
Returns a generalized morphism with source aid by three arrows or by cospan, or a generalized morphism by span, depending on the standard.
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