chap1.txt

  
  1 Generalized Morphism Category
  
  Let  \mathbf{A}  be  an abelian category. We denote its generalized morphism
  category by \mathbf{G(A)}.
  
  
  1.1 GAP Categories
  
  1.1-1 IsGeneralizedMorphismCategoryObject
  
  IsGeneralizedMorphismCategoryObject( object )  filter
  Returns:  true or false
  
  The GAP category of objects in the generalized morphism category.
  
  1.1-2 IsGeneralizedMorphism
  
  IsGeneralizedMorphism( object )  filter
  Returns:  true or false
  
  The GAP category of morphisms in the generalized morphism category.
  
  
  1.2 Attributes
  
  1.2-1 UnderlyingHonestObject
  
  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
  
  1.2-2 DomainOfGeneralizedMorphism
  
  DomainOfGeneralizedMorphism( 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}.
  
  1.2-3 Codomain
  
  Codomain( 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}.
  
  1.2-4 AssociatedMorphism
  
  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}.
  
  1.2-5 DomainAssociatedMorphismCodomainTriple
  
  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.
  
  1.2-6 HonestRepresentative
  
  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.
  
  1.2-7 GeneralizedInverse
  
  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.
  
  1.2-8 IdempotentDefinedBySubobject
  
  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.
  
  1.2-9 IdempotentDefinedByFactorobject
  
  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.
  
  1.2-10 UnderlyingHonestCategory
  
  UnderlyingHonestCategory( C )  attribute
  Returns:  a category
  
  The  argument  is  a  generalized  morphism  category C = \mathbf{G(A)}. The
  output is \mathbf{A}.
  
  
  1.3 Operations
  
  1.3-1 GeneralizedMorphismFromFactorToSubobject
  
  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.
  
  1.3-2 CommonRestriction
  
  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.
  
  1.3-3 ConcatenationProduct
  
  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.
  
  
  1.4 Properties
  
  1.4-1 IsHonest
  
  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.
  
  1.4-2 HasFullDomain
  
  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.
  
  1.4-3 HasFullCodomain
  
  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.
  
  1.4-4 IsSingleValued
  
  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.
  
  1.4-5 IsTotal
  
  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.
  
  
  1.5 Convenience methods
  
  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.
  
  1.5-1 GeneralizedMorphismCategory
  
  GeneralizedMorphismCategory( C )  operation
  Returns:  a category
  
  Creates   a   new   category   of  generalized  morphisms.  Might  point  to
  GeneralizedMorphismCategoryByThreeArrows,
  GeneralizedMorphismCategoryByCospans, or GeneralizedMorphismCategoryBySpans
  
  1.5-2 GeneralizedMorphismObject
  
  GeneralizedMorphismObject( A )  operation
  Returns:  an object in the generalized morphism category
  
  Creates an object in the current generalized morphism category, depending on
  the standard
  
  1.5-3 AsGeneralizedMorphism
  
  AsGeneralizedMorphism( phi )  operation
  Returns:  a generalized morphism
  
  Returns  the  corresponding  morphism  to  phi  in  the  current generalized
  morphism category.
  
  1.5-4 GeneralizedMorphism
  
  GeneralizedMorphism( phi, psi )  operation
  Returns:  a generalized morphism
  
  Returns the corresponding morphism to phi and psi in the current generalized
  morphism category.
  
  1.5-5 GeneralizedMorphism
  
  GeneralizedMorphism( iota, phi, pi )  operation
  Returns:  a generalized morphism
  
  Returns  the  corresponding  morphism  to  iota,  phi and psi in the current
  generalized morphism category.
  
  1.5-6 GeneralizedMorphismWithRangeAid
  
  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.
  
  1.5-7 GeneralizedMorphismWithSourceAid
  
  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.