chap2.txt

  
  2 Category of Categories
  
  Categories  itself  with  functors as morphisms form a category. So the data
  structure  of  CapCategorys  is  designed  to be objects in a category. This
  category  is  implemented  in  CapCat. For every category, the corresponding
  object  in  Cat  can  be  obtained via AsCatObject. The implemetation of the
  category  of  categories  offers  a  data  structure for functors. Those are
  implemented  as  morphisms in this category, so functors can be handled like
  morphisms  in  a category. Also convenience functions to install functors as
  methods are implemented (in order to avoid ApplyFunctor).
  
  
  2.1 The Category Cat
  
  2.1-1 CapCat
  
  CapCat global variable
  
  This  variable  stores  the category of categories. Every category object is
  constructed  as  an  object  in  this  category,  so Cat is constructed when
  loading the package.
  
  
  2.2 Categories
  
  2.2-1 IsCapCategoryAsCatObject
  
  IsCapCategoryAsCatObject( object )  filter
  Returns:  true or false
  
  The GAP category of CAP categories seen as object in Cat.
  
  2.2-2 IsCapFunctor
  
  IsCapFunctor( object )  filter
  Returns:  true or false
  
  The GAP category of functors.
  
  2.2-3 IsCapNaturalTransformation
  
  IsCapNaturalTransformation( object )  filter
  Returns:  true or false
  
  The GAP category of natural transformations.
  
  
  2.3 Constructors
  
  2.3-1 AsCatObject
  
  AsCatObject( C )  attribute
  
  Given a CAP category C, this method returns the corresponding object in Cat.
  For  technical  reasons,  the filter IsCapCategory must not imply the filter
  IsCapCategoryObject.  For example, if InitialObject is applied to an object,
  it  returns  the  initial  object  of  its  category.  If it is applied to a
  category,  it  returns the initial object of the category. If a CAP category
  would  be  a  category object itself, this would be ambiguous. So categories
  must  be  wrapped in a CatObject to be an object in Cat. This method returns
  the wrapper object. The category can be reobtained by AsCapCategory.
  
  2.3-2 AsCapCategory
  
  AsCapCategory( C )  attribute
  
  For  an  object  C  in Cat, this method returns the underlying CAP category.
  This  method is inverse to AsCatObject, i.e. AsCapCategory( AsCatObject( A )
  ) = A.
  
  
  2.4 Functors
  
  Functors  are  morphisms  in Cat, thus they have source and target which are
  categories. A multivariate functor can be constructed via a product category
  as  source,  a  presheaf is constructed via the opposite category as source.
  Moreover,  an  object  and a morphism function can be added to a functor, to
  apply it to objects or morphisms in the source category.
  
  2.4-1 CapFunctor
  
  CapFunctor( name, A, B )  operation
  CapFunctor( name, A, B )  operation
  CapFunctor( name, A, B )  operation
  CapFunctor( name, A, B )  operation
  CapFunctor( name, A, B )  operation
  CapFunctor( name, A, B )  operation
  
  These  methods  construct a CAP functor, i.e. a morphism in Cat. Name should
  be  an  unique  name  for  the  functor, it is also used when the functor is
  installed  as  a method. A and B are source and target. Both can be given as
  object in Cat or as category itself.
  
  2.4-2 AddObjectFunction
  
  AddObjectFunction( functor, function )  operation
  
  This  operation  adds a function to the functor which can then be applied to
  objects  in the source. The given function function has to take one argument
  which  must  be  an  object  in  the  source  category  and  should return a
  CapCategoryObject.  The  object  is  automatically added to the range of the
  functor when it is applied to the object.
  
  2.4-3 FunctorObjectOperation
  
  FunctorObjectOperation( F )  attribute
  Returns:  a GAP operation
  
  The  argument  is a functor F. The output is the GAP operation realizing the
  action of F on objects.
  
  2.4-4 AddMorphismFunction
  
  AddMorphismFunction( functor, function )  operation
  
  This  operation  adds a function to the functor which can then be applied to
  morphisms  in  the  source.  The  given  function function has to take three
  arguments  A,  \tau,  B.  When the funtor functor is applied to the morphism
  \tau,  A  is  the  result of functor applied to the source of \tau, B is the
  result of functor applied to the range.
  
  2.4-5 FunctorMorphismOperation
  
  FunctorMorphismOperation( F )  attribute
  Returns:  a GAP operation
  
  The  argument  is a functor F. The output is the GAP operation realizing the
  action of F on morphisms.
  
  2.4-6 ApplyFunctor
  
  ApplyFunctor( func, A )  function
  Returns:  IsCapCategoryCell
  
  Applies the functor func to the object or morphism A.
  
  2.4-7 InstallFunctor
  
  InstallFunctor( functor, method_name )  operation
  
  TODO
  
  2.4-8 IdentityFunctor
  
  IdentityFunctor( category )  attribute
  Returns:  a functor
  
  Returns  the identity functor of the category cat viewed as an object in the
  category of categories.
  
  2.4-9 FunctorCanonicalizeZeroObjects
  
  FunctorCanonicalizeZeroObjects( category )  attribute
  Returns:  a functor
  
  Returns  the  endofunctor  of  the  category  cat  with zero which maps each
  (object  isomorphic  to  the)  zero  object to ZeroObject(cat) and to itself
  otherwise. This functor is equivalent to the identity functor.
  
  2.4-10 NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects
  
  NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects( category )  attribute
  Returns:  a natural transformation
  
  Returns   the   natural   isomorphism   from   the   identity   functor   to
  FunctorCanonicalizeZeroObjects(cat).
  
  2.4-11 FunctorCanonicalizeZeroMorphisms
  
  FunctorCanonicalizeZeroMorphisms( category )  attribute
  Returns:  a functor
  
  Returns the endofunctor of the category cat with zero which maps each object
  to  itself,  each  morphism  ϕ to itself, unless it is congruent to the zero
  morphism;  in  this  case it is mapped to ZeroMorphism(Source(ϕ), Range(ϕ)).
  This functor is equivalent to the identity functor.
  
  2.4-12 NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms
  
  NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms( category )  attribute
  Returns:  a natural transformation
  
  Returns   the   natural   isomorphism   from   the   identity   functor   to
  FunctorCanonicalizeZeroMorphisms(cat).
  
  
  2.5 Natural transformations
  
  2.5-1 Name
  
  Name( arg )  attribute
  Returns:  a string
  
  As  every functor, every natural transformation has a name attribute. It has
  to be a string and will be set by the Constructor.
  
  2.5-2 NaturalTransformation
  
  NaturalTransformation( [name, ]F, G )  operation
  Returns:  a natural transformation
  
  Constructs  a  natural transformation between the functors F:A \rightarrow B
  and  G:A  \rightarrow B. The string name is optional, and, if not given, set
  automatically from the names of the functors
  
  2.5-3 AddNaturalTransformationFunction
  
  AddNaturalTransformationFunction( N, func )  operation
  
  Adds  the function (or list of functions) func to the natural transformation
  N. The function or each function in the list should take three arguments. If
  N:  F  \rightarrow  G,  the  arguments  should be F(A), A, G(A). The ouptput
  should be a morphism, F(A) \rightarrow G(A).
  
  2.5-4 ApplyNaturalTransformation
  
  ApplyNaturalTransformation( N, A )  function
  Returns:  a morphism
  
  Given  a  natural  transformation  N:F  \rightarrow  G and an object A, this
  function  should return the morphism F(A) \rightarrow G(A), corresponding to
  N.
  
  2.5-5 InstallNaturalTransformation
  
  InstallNaturalTransformation( N, name )  operation
  
  Installs  the  natural  transformation  N  as  operation with the name name.
  Argument for this operation is an object, output is a morphism.
  
  2.5-6 HorizontalPreComposeNaturalTransformationWithFunctor
  
  HorizontalPreComposeNaturalTransformationWithFunctor( N, F )  operation
  Returns:  a natural transformation
  
  Computes the horizontal composition of the natural transformation N and
  
  2.5-7 HorizontalPreComposeFunctorWithNaturalTransformation
  
  HorizontalPreComposeFunctorWithNaturalTransformation( F, N )  operation
  Returns:  a natural transformation
  
  Computes  the  horizontal  composition  of  the  functor  F  and the natural
  transformation N.