chap4.txt

  
  4 Generalized Morphism Category by Three Arrows
  
  
  4.1 GAP Categories
  
  4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject
  
  IsGeneralizedMorphismCategoryByThreeArrowsObject( object )  filter
  Returns:  true or false
  
  The  GAP  category  of objects in the generalized morphism category by three
  arrows.
  
  4.1-2 IsGeneralizedMorphismByThreeArrows
  
  IsGeneralizedMorphismByThreeArrows( object )  filter
  Returns:  true or false
  
  The  GAP category of morphisms in the generalized morphism category by three
  arrows.
  
  
  4.2 Properties
  
  4.2-1 HasIdentitiesAsReversedArrows
  
  HasIdentitiesAsReversedArrows( alpha )  property
  Returns:  true or false
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow  c  \leftarrow  d.  The  output  is true if a \leftarrow b and c
  \leftarrow d are congruent to identity morphisms, false otherwise.
  
  4.2-2 HasIdentityAsSourceAid
  
  HasIdentityAsSourceAid( alpha )  property
  Returns:  true or false
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow  c  \leftarrow  d.  The  output  is  true  if  a \leftarrow b is
  congruent to an identity morphism, false otherwise.
  
  4.2-3 HasIdentityAsRangeAid
  
  HasIdentityAsRangeAid( alpha )  property
  Returns:  true or false
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow  c  \leftarrow  d.  The  output  is  true  if  c \leftarrow d is
  congruent to an identity morphism, false otherwise.
  
  
  4.3 Attributes
  
  4.3-1 UnderlyingHonestObject
  
  UnderlyingHonestObject( a )  attribute
  Returns:  an object in \mathbf{A}
  
  The  argument  is  an object a in the generalized morphism category by three
  arrows. The output is its underlying honest object.
  
  4.3-2 SourceAid
  
  SourceAid( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(b,a)
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow c \leftarrow d. The output is its source aid a \leftarrow b.
  
  4.3-3 RangeAid
  
  RangeAid( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(d,c)
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow c \leftarrow d. The output is its range aid c \leftarrow d.
  
  4.3-4 Arrow
  
  Arrow( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(b,c)
  
  The argument is a generalized morphism \alpha by three arrows a \leftarrow b
  \rightarrow c \leftarrow d. The output is its range aid b \rightarrow c.
  
  4.3-5 PseudoInverse
  
  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 three
  arrows. The output is its pseudo inverse b \rightarrow a.
  
  4.3-6 GeneralizedInverseByThreeArrows
  
  GeneralizedInverseByThreeArrows( 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 three arrows.
  
  4.3-7 IdempotentDefinedBySubobjectByThreeArrows
  
  IdempotentDefinedBySubobjectByThreeArrows( 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 three arrows
  defined by \alpha.
  
  4.3-8 IdempotentDefinedByFactorobjectByThreeArrows
  
  IdempotentDefinedByFactorobjectByThreeArrows( 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 three arrows defined by \alpha.
  
  
  4.4 Operations
  
  4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows
  
  GeneralizedMorphismFromFactorToSubobjectByThreeArrows( 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 three arrows from the factorobject to the subobject.
  
  4.4-2 CommonCoastriction
  
  CommonCoastriction( L )  operation
  Returns:  a list of generalized morphisms
  
  The argument is a list L of generalized morphisms by three arrows having the
  same  range.  The  output is a list of generalized morphisms by three arrows
  which is the comman coastriction of L.
  
  
  4.5 Constructors
  
  4.5-1 GeneralizedMorphismByThreeArrows
  
  GeneralizedMorphismByThreeArrows( 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 three arrows with source aid \alpha, arrow \beta, and range aid
  \gamma.
  
  4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid
  
  GeneralizedMorphismByThreeArrowsWithSourceAid( 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 three arrows
  defined  by  the  composition  the  given two arrows regarded as generalized
  morphisms.
  
  4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid
  
  GeneralizedMorphismByThreeArrowsWithRangeAid( beta, gamma )  operation
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,d)
  
  The arguments are morphisms \beta: b \rightarrow c, and \gamma: c \leftarrow
  d  in  \mathbf{A}.  The  output  is  a  generalized morphism by three arrows
  defined  by  the  composition  the  given two arrows regarded as generalized
  morphisms.
  
  4.5-4 AsGeneralizedMorphismByThreeArrows
  
  AsGeneralizedMorphismByThreeArrows( 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 three arrows defined by \alpha.
  
  4.5-5 GeneralizedMorphismCategoryByThreeArrows
  
  GeneralizedMorphismCategoryByThreeArrows( A )  attribute
  Returns:  a category
  
  The   argument  is  an  abelian  category  \mathbf{A}.  The  output  is  its
  generalized morphism category \mathbf{G(A)} by three arrows.
  
  4.5-6 GeneralizedMorphismByThreeArrowsObject
  
  GeneralizedMorphismByThreeArrowsObject( 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 three arrows whose
  underlying honest object is a.