chap2.txt

  
  2 Generalized Morphism Category by Cospans
  
  
  2.1 GAP Categories
  
  2.1-1 IsGeneralizedMorphismCategoryByCospansObject
  
  IsGeneralizedMorphismCategoryByCospansObject( object )  filter
  Returns:  true or false
  
  The GAP category of objects in the generalized morphism category by cospans.
  
  2.1-2 IsGeneralizedMorphismByCospan
  
  IsGeneralizedMorphismByCospan( object )  filter
  Returns:  true or false
  
  The  GAP  category  of  morphisms  in  the  generalized morphism category by
  cospans.
  
  
  2.2 Properties
  
  2.2-1 HasIdentityAsReversedArrow
  
  HasIdentityAsReversedArrow( alpha )  property
  Returns:  true or false
  
  The  argument  is  a generalized morphism \alpha by a cospan a \rightarrow b
  \leftarrow  c.  The  output  is  true  if  b \leftarrow c is congruent to an
  identity morphism, false otherwise.
  
  
  2.3 Attributes
  
  2.3-1 UnderlyingHonestObject
  
  UnderlyingHonestObject( a )  attribute
  Returns:  an object in \mathbf{A}
  
  The argument is an object a in the generalized morphism category by cospans.
  The output is its underlying honest object.
  
  2.3-2 Arrow
  
  Arrow( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(a,c)
  
  The  argument  is  a generalized morphism \alpha by a cospan a \rightarrow b
  \leftarrow c. The output is its arrow a \rightarrow b.
  
  2.3-3 ReversedArrow
  
  ReversedArrow( alpha )  attribute
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{A}}(c,b)
  
  The  argument  is  a generalized morphism \alpha by a cospan a \rightarrow b
  \leftarrow c. The output is its reversed arrow b \leftarrow c.
  
  2.3-4 NormalizedCospanTuple
  
  NormalizedCospanTuple( alpha )  attribute
  Returns:  a pair of morphisms in \mathbf{A}.
  
  The  argument is a generalized morphism \alpha: a \rightarrow b by a cospan.
  The output is its normalized cospan pair (a \rightarrow d, d \leftarrow b).
  
  2.3-5 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 cospan.
  The output is its pseudo inverse b \rightarrow a.
  
  2.3-6 GeneralizedInverseByCospan
  
  GeneralizedInverseByCospan( 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 cospan.
  
  2.3-7 IdempotentDefinedBySubobjectByCospan
  
  IdempotentDefinedBySubobjectByCospan( 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 cospan defined
  by \alpha.
  
  2.3-8 IdempotentDefinedByFactorobjectByCospan
  
  IdempotentDefinedByFactorobjectByCospan( 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 cospan defined by \alpha.
  
  2.3-9 NormalizedCospan
  
  NormalizedCospan( 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 cospan.
  The output is its normalization by cospan.
  
  
  2.4 Operations
  
  2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan
  
  GeneralizedMorphismFromFactorToSubobjectByCospan( 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 cospan from the factorobject to the subobject.
  
  
  2.5 Constructors
  
  2.5-1 GeneralizedMorphismByCospan
  
  GeneralizedMorphismByCospan( alpha, beta )  operation
  Returns:  a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c)
  
  The arguments are morphisms \alpha: a \rightarrow b and \beta: c \rightarrow
  b  in  \mathbf{A}. The output is a generalized morphism by cospan with arrow
  \alpha and reversed arrow \beta.
  
  2.5-2 GeneralizedMorphismByCospan
  
  GeneralizedMorphismByCospan( 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  cospan  defined  by  the  composition  the  given three arrows
  regarded as generalized morphisms.
  
  2.5-3 GeneralizedMorphismByCospanWithSourceAid
  
  GeneralizedMorphismByCospanWithSourceAid( 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 cospan defined by
  the composition the given two arrows regarded as generalized morphisms.
  
  2.5-4 AsGeneralizedMorphismByCospan
  
  AsGeneralizedMorphismByCospan( 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 cospan defined by \alpha.
  
  2.5-5 GeneralizedMorphismCategoryByCospans
  
  GeneralizedMorphismCategoryByCospans( A )  attribute
  Returns:  a category
  
  The   argument  is  an  abelian  category  \mathbf{A}.  The  output  is  its
  generalized morphism category \mathbf{G(A)} by cospans.
  
  2.5-6 GeneralizedMorphismByCospansObject
  
  GeneralizedMorphismByCospansObject( 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 cospans whose underlying
  honest object is a.
  
  
  2.6 Constructors of lifts of exact functors and natrual (iso)morphisms
  
  2.6-1 AsGeneralizedMorphismByCospan
  
  AsGeneralizedMorphismByCospan( F, name )  operation
  
  Lift   the   exact   functor   F   to   a   functor  A  ->  B,  where  A  :=
  GeneralizedMorphismCategoryByCospans(  AsCapCategory(  Source( F ) ) ) and B
  := GeneralizedMorphismCategoryByCospans( AsCapCategory( Range( F ) ) ).