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  1 Generalized Morphism Category
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  Let  \mathbf{A}  be  an abelian category. We denote its generalized morphism
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  category by \mathbf{G(A)}.
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  1.1 GAP Categories
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  1.1-1 IsGeneralizedMorphismCategoryObject
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  IsGeneralizedMorphismCategoryObject( object )  filter
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  Returns:  true or false
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  The GAP category of objects in the generalized morphism category.
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  1.1-2 IsGeneralizedMorphism
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  IsGeneralizedMorphism( object )  filter
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  Returns:  true or false
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  The GAP category of morphisms in the generalized morphism category.
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  1.2 Attributes
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  1.2-1 UnderlyingHonestObject
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  UnderlyingHonestObject( a )  attribute
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  Returns:  an object in \mathbf{A}
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  The argument is an object a in the generalized morphism category. The output
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  is its underlying honest object
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  1.2-2 DomainOfGeneralizedMorphism
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  DomainOfGeneralizedMorphism( alpha )  attribute
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}( d, a )
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  The  argument  is a generalized morphism \alpha: a \rightarrow b. The output
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  is its domain d \hookrightarrow a \in \mathbf{A}.
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  1.2-3 Codomain
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  Codomain( alpha )  attribute
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}( b, c )
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  The  argument  is a generalized morphism \alpha: a \rightarrow b. The output
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  is its codomain b \twoheadrightarrow c \in \mathbf{A}.
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  1.2-4 AssociatedMorphism
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  AssociatedMorphism( alpha )  attribute
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}( d, c )
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  The  argument  is a generalized morphism \alpha: a \rightarrow b. The output
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  is its associated morphism d \rightarrow c \in \mathbf{A}.
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  1.2-5 DomainAssociatedMorphismCodomainTriple
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  DomainAssociatedMorphismCodomainTriple( alpha )  attribute
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  Returns:  a triple of morphisms in \mathbf{A}
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  The  argument  is a generalized morphism \alpha: a \rightarrow b. The output
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  is a triple ( d \hookrightarrow a, d \rightarrow c, b \twoheadrightarrow c )
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  consisting of its domain, associated morphism, and codomain.
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  1.2-6 HonestRepresentative
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  HonestRepresentative( alpha )  attribute
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}( a, b )
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  The  argument  is a generalized morphism \alpha: a \rightarrow b. The output
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  is  the  honest  representative  in  \mathbf{A}  of  \alpha,  if  it exists,
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  otherwise an error is thrown.
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  1.2-7 GeneralizedInverse
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  GeneralizedInverse( alpha )  operation
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a)
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  The  argument  is  a  morphism  \alpha:  a \rightarrow b \in \mathbf{A}. The
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  output is its generalized inverse b \rightarrow a.
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  1.2-8 IdempotentDefinedBySubobject
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  IdempotentDefinedBySubobject( alpha )  operation
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b)
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  The  argument is a subobject \alpha: a \hookrightarrow b \in \mathbf{A}. The
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  output  is  the  idempotent  b  \rightarrow  b  \in \mathbf{G(A)} defined by
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  \alpha.
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  1.2-9 IdempotentDefinedByFactorobject
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  IdempotentDefinedByFactorobject( alpha )  operation
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b)
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  The   argument   is  a  factorobject  \alpha:  b  \twoheadrightarrow  a  \in
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  \mathbf{A}.  The  output is the idempotent b \rightarrow b \in \mathbf{G(A)}
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  defined by \alpha.
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  1.2-10 UnderlyingHonestCategory
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  UnderlyingHonestCategory( C )  attribute
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  Returns:  a category
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  The  argument  is  a  generalized  morphism  category C = \mathbf{G(A)}. The
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  output is \mathbf{A}.
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  1.3 Operations
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  1.3-1 GeneralizedMorphismFromFactorToSubobject
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  GeneralizedMorphismFromFactorToSubobject( beta, alpha )  operation
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  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(c,a)
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  The  arguments  are  a  a  factorobject \beta: b \twoheadrightarrow c, and a
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  subobject  \alpha:  a  \hookrightarrow  b.  The  output  is  the generalized
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  morphism from the factorobject to the subobject.
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  1.3-2 CommonRestriction
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  CommonRestriction( L )  operation
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  Returns:  a list of generalized morphisms
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  The argument is a list L of generalized morphisms by three arrows having the
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  same  source.  The output is a list of generalized morphisms by three arrows
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  which is the comman restriction of L.
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  1.3-3 ConcatenationProduct
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  ConcatenationProduct( L )  operation
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  Returns:  a generalized moprhism
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  The  argument  is  a  list  L = ( \alpha_1, \dots, \alpha_n ) of generalized
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  morphisms  (with  same  data  structures). The output is their concatenation
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  product,      i.e.,      a      generalized     morphism     \alpha     with
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  \mathrm{UnderlyingHonestObject}(    \mathrm{Source}(    \alpha    )    )   =
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  \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha_i
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  )  ),  and  \mathrm{UnderlyingHonestObject}(  \mathrm{Range}(  \alpha  ) ) =
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  \bigoplus_{i=1}^n  \mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha_i
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  )  ), and with morphisms in the representation of \alpha given as the direct
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  sums of the corresponding morphisms of the \alpha_i.
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  1.4 Properties
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  1.4-1 IsHonest
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  IsHonest( alpha )  property
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  Returns:  a boolean
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  The  argument is a generalized morphism \alpha. The output is true if \alpha
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  admits an honest representative, otherwise false.
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  1.4-2 HasFullDomain
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  HasFullDomain( alpha )  property
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  Returns:  a boolean
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  The  argument  is  a  generalized morphism \alpha. The output is true if the
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  domain of \alpha is an isomorphism, otherwise false.
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  1.4-3 HasFullCodomain
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  HasFullCodomain( alpha )  property
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  Returns:  a boolean
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  The  argument  is  a  generalized morphism \alpha. The output is true if the
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  codomain of \alpha is an isomorphism, otherwise false.
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  1.4-4 IsSingleValued
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  IsSingleValued( alpha )  property
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  Returns:  a boolean
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  The  argument  is  a  generalized morphism \alpha. The output is true if the
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  codomain of \alpha is an isomorphism, otherwise false.
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  1.4-5 IsTotal
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  IsTotal( alpha )  property
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  Returns:  a boolean
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  The  argument  is  a  generalized morphism \alpha. The output is true if the
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  domain of \alpha is an isomorphism, otherwise false.
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  1.5 Convenience methods
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  This section contains operations which, depending on the current generalized
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  morphism  standard  of  the  system  and  the category, might point to other
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  Operations. Please use them only as convenience and never in serious code.
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  1.5-1 GeneralizedMorphismCategory
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  GeneralizedMorphismCategory( C )  operation
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  Returns:  a category
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  Creates   a   new   category   of  generalized  morphisms.  Might  point  to
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  GeneralizedMorphismCategoryByThreeArrows,
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  GeneralizedMorphismCategoryByCospans, or GeneralizedMorphismCategoryBySpans
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  1.5-2 GeneralizedMorphismObject
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  GeneralizedMorphismObject( A )  operation
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  Returns:  an object in the generalized morphism category
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  Creates an object in the current generalized morphism category, depending on
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  the standard
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  1.5-3 AsGeneralizedMorphism
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  AsGeneralizedMorphism( phi )  operation
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  Returns:  a generalized morphism
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  Returns  the  corresponding  morphism  to  phi  in  the  current generalized
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  morphism category.
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  1.5-4 GeneralizedMorphism
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  GeneralizedMorphism( phi, psi )  operation
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  Returns:  a generalized morphism
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  Returns the corresponding morphism to phi and psi in the current generalized
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  morphism category.
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  1.5-5 GeneralizedMorphism
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  GeneralizedMorphism( iota, phi, pi )  operation
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  Returns:  a generalized morphism
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  Returns  the  corresponding  morphism  to  iota,  phi and psi in the current
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  generalized morphism category.
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  1.5-6 GeneralizedMorphismWithRangeAid
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  GeneralizedMorphismWithRangeAid( arg1, arg2 )  operation
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  Returns a generalized morphism with range aid by three arrows or by span, or
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  a generalized morphism by cospan, depending on the standard.
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  1.5-7 GeneralizedMorphismWithSourceAid
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  GeneralizedMorphismWithSourceAid( arg1, arg2 )  operation
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  Returns a generalized morphism with source aid by three arrows or by cospan,
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  or a generalized morphism by span, depending on the standard.
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