chap1.txt

  
  1 CAP Categories
  
  Categories  are  the main GAP objects in CAP. They are used to associate GAP
  objects  which  represent  objects  and  morphisms  with  their category. By
  associating  a  GAP  object to the category, one of two filters belonging to
  the category (ObjectFilter/MorphismFilter) are set to true. Via Add methods,
  functions  for  specific  existential  quantifiers  can be associated to the
  category and after that can be applied to GAP objects in the category. A GAP
  category  object  also  knows  which constructions are currently possible in
  this category.
  
  
  1.1 Categories
  
  1.1-1 IsCapCategory
  
  IsCapCategory( object )  filter
  Returns:  true or false
  
  The  GAP  category  of  CAP  categories. Objects of this type handle the CAP
  category  information,  the  caching,  and  filters  for  objects in the CAP
  category.  Please note that the object itself is not related to methods, you
  only need it as a handler and a presentation of the CAP category.
  
  1.1-2 IsCapCategoryCell
  
  IsCapCategoryCell( object )  filter
  Returns:  true or false
  
  The  GAP  category of CAP category cells. Every object, morphism, and 2-cell
  of a CAP category lies in this GAP category.
  
  1.1-3 IsCapCategoryObject
  
  IsCapCategoryObject( object )  filter
  Returns:  true or false
  
  The  GAP  category  of  CAP category objects. Every object of a CAP category
  lies in this GAP category.
  
  1.1-4 IsCapCategoryMorphism
  
  IsCapCategoryMorphism( object )  filter
  Returns:  true or false
  
  The GAP category of CAP category morphisms. Every morphism of a CAP category
  lies in this GAP category.
  
  1.1-5 IsCapCategoryTwoCell
  
  IsCapCategoryTwoCell( object )  filter
  Returns:  true or false
  
  The  GAP  category  of  CAP category 2-cells. Every 2-cell of a CAP category
  lies in this GAP category.
  
  
  1.2 Constructor
  
  1.2-1 CreateCapCategory
  
  CreateCapCategory(  )  operation
  Returns:  a category
  
  Creates a new CAP category from scratch. It gets a generic name.
  
  1.2-2 CreateCapCategory
  
  CreateCapCategory( s )  operation
  Returns:  a category
  
  The  argument  is a string s. This operation creates a new CAP category from
  scratch. Its name is set to s.
  
  
  1.3 Internal Attributes
  
  Each  category  C  stores  various filters. They are used to apply the right
  functions in the method selection.
  
  1.3-1 CategoryFilter
  
  CategoryFilter( C )  attribute
  Returns:  a filter
  
  The argument is a cateogry C. The output is a filter in which C lies.
  
  1.3-2 CellFilter
  
  CellFilter( C )  attribute
  Returns:  a filter
  
  The argument is a cateogry C. The output is a filter in which all cells of C
  shall lie.
  
  1.3-3 ObjectFilter
  
  ObjectFilter( C )  attribute
  Returns:  a filter
  
  The argument is a cateogry C. The output is a filter in which all objects of
  C shall lie.
  
  1.3-4 MorphismFilter
  
  MorphismFilter( C )  attribute
  Returns:  a filter
  
  The  argument is a cateogry C. The output is a filter in which all morphisms
  of C shall lie.
  
  1.3-5 TwoCellFilter
  
  TwoCellFilter( C )  attribute
  Returns:  a filter
  
  The argument is a cateogry C. The output is a filter in which all 2-cells of
  C shall lie.
  
  
  1.4 Logic switcher
  
  1.4-1 CapCategorySwitchLogicOn
  
  CapCategorySwitchLogicOn( C )  function
  
  Activates the predicate implication logic for the category C.
  
  1.4-2 CapCategorySwitchLogicOff
  
  CapCategorySwitchLogicOff( C )  function
  
  Deactivates the predicate implication logic for the category C.
  
  
  1.5 Tool functions
  
  1.5-1 CanCompute
  
  CanCompute( C, s )  operation
  Returns:  true or false
  
  The  argument  is a category C and a string s, which should be the name of a
  primitive  operation,  e.g., PreCompose. If applying this method is possible
  in  C,  the  method  returns true, false otherwise. If the string is not the
  name of a primitive operation, an error is raised.
  
  1.5-2 CheckConstructivenessOfCategory
  
  CheckConstructivenessOfCategory( C, s )  operation
  Returns:  a list
  
  The  arguments  are  a  category  C  and  a  string s. If s is a categorical
  property  (e.g.  "IsAbelianCategory"),  the output is a list of strings with
  basic  operations  which  are  missing in C to have the categorical property
  constructively. If s is not a categorical property, an error is raised.
  
  
  1.6 Well-Definedness of Cells
  
  1.6-1 IsWellDefined
  
  IsWellDefined( c )  property
  Returns:  a boolean
  
  The argument is a cell c. The output is true if c is well-defined, otherwise
  the output is false.
  
  
  1.7 Type check
  
  1.7-1 DisableBasicOperationTypeCheck
  
  DisableBasicOperationTypeCheck( category )  function
  EnableBasicOperationTypeCheck( arg )  function
  
  Most  basic  operations have a prefunction, which does a (sometimes partial)
  typecheck  at  the  beginning  of  the  operation. These functions enable or
  disable this check for a category. (Enabled by default)