chap3.txt

  
  3 Generalized Morphism Category by Spans
  
  
  3.1 GAP Categories
  
  3.1-1 IsGeneralizedMorphismCategoryBySpansObject
  
  IsGeneralizedMorphismCategoryBySpansObject( object )  filter
  Returns:  true or false
  
  The GAP category of objects in the generalized morphism category by spans.
  
  3.1-2 IsGeneralizedMorphismBySpan
  
  IsGeneralizedMorphismBySpan( object )  filter
  Returns:  true or false
  
  The GAP category of morphisms in the generalized morphism category by spans.
  
  
  3.2 Properties
  
  3.2-1 HasIdentityAsReversedArrow
  
  HasIdentityAsReversedArrow( alpha )  property
  Returns:  true or false
  
  The  argument  is  a  generalized  morphism  \alpha by a span a \leftarrow b
  \rightarrow  c.  The  output  is  true  if a \leftarrow b is congruent to an
  identity morphism, false otherwise.
  
  
  3.3 Attributes
  
  3.3-1 UnderlyingHonestObject
  
  UnderlyingHonestObject( a )  attribute
  Returns:  an object in \mathbf{A}
  
  The  argument  is an object a in the generalized morphism category by spans.
  The output is its underlying honest object.
  
  3.3-2 Arrow
  
  Arrow( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(b,c)
  
  The  argument  is  a  generalized  morphism  \alpha by a span a \leftarrow b
  \rightarrow c. The output is its arrow b \rightarrow c.
  
  3.3-3 ReversedArrow
  
  ReversedArrow( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(b,a)
  
  The  argument  is  a  generalized  morphism  \alpha by a span a \leftarrow b
  \rightarrow c. The output is its reversed arrow a \leftarrow b.
  
  3.3-4 NormalizedSpanTuple
  
  NormalizedSpanTuple( alpha )  attribute
  Returns:  a pair of morphisms in \mathbf{A}.
  
  The  argument  is  a generalized morphism \alpha: a \rightarrow b by a span.
  The output is its normalized span pair (a \leftarrow d, d \rightarrow b).
  
  3.3-5 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 a span.
  The output is its pseudo inverse b \rightarrow a.
  
  3.3-6 GeneralizedInverseBySpan
  
  GeneralizedInverseBySpan( 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 span.
  
  3.3-7 IdempotentDefinedBySubobjectBySpan
  
  IdempotentDefinedBySubobjectBySpan( 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 span defined
  by \alpha.
  
  3.3-8 IdempotentDefinedByFactorobjectBySpan
  
  IdempotentDefinedByFactorobjectBySpan( 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 span defined by \alpha.
  
  3.3-9 NormalizedSpan
  
  NormalizedSpan( 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 span.
  The output is its normalization by span.
  
  
  3.4 Operations
  
  3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan
  
  GeneralizedMorphismFromFactorToSubobjectBySpan( 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 span from the factorobject to the subobject.
  
  
  3.5 Constructors
  
  3.5-1 GeneralizedMorphismBySpan
  
  GeneralizedMorphismBySpan( alpha, beta )  operation
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,b)
  
  The  arguments are morphisms \alpha: a \leftarrow c and \beta: c \rightarrow
  b  in  \mathbf{A}.  The  output is a generalized morphism by span with arrow
  \beta and reversed arrow \alpha.
  
  3.5-2 GeneralizedMorphismBySpan
  
  GeneralizedMorphismBySpan( 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 span defined by the composition the given three arrows regarded
  as generalized morphisms.
  
  3.5-3 GeneralizedMorphismBySpanWithRangeAid
  
  GeneralizedMorphismBySpanWithRangeAid( alpha, beta )  operation
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c)
  
  The arguments are morphisms \alpha: a \rightarrow b, and \beta: b \leftarrow
  c in \mathbf{A}. The output is a generalized morphism by span defined by the
  composition the given two arrows regarded as generalized morphisms.
  
  3.5-4 AsGeneralizedMorphismBySpan
  
  AsGeneralizedMorphismBySpan( 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 span defined by \alpha.
  
  3.5-5 GeneralizedMorphismCategoryBySpans
  
  GeneralizedMorphismCategoryBySpans( A )  attribute
  Returns:  a category
  
  The   argument  is  an  abelian  category  \mathbf{A}.  The  output  is  its
  generalized morphism category \mathbf{G(A)} by spans.
  
  3.5-6 GeneralizedMorphismBySpansObject
  
  GeneralizedMorphismBySpansObject( 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 spans whose underlying
  honest object is a.