10 Technical Details
  
  
  10.1 The Category Cat
  
  10.1-1 ObjectCache
  
  ObjectCache( functor )  attribute
  Returns:  IsCachingObject
  
  Retuns  the  caching  object which stores the results of the functor functor
  applied to objects.
  
  10.1-2 MorphismCache
  
  MorphismCache( functor )  attribute
  Returns:  IsCachingObject
  
  Retuns  the  caching  object which stores the results of the functor functor
  applied to morphisms.
  
  
  10.2 Install Functions for IsWellDefined
  
  10.2-1 InstallMethodWithToDoForIsWellDefined
  
  InstallMethodWithToDoForIsWellDefined( arg )  function
  
  The  IsWellDefined  filter  is a basic function of CAP. For every categorial
  construction  the  outcome is well defined if and only if every input object
  or morphism of the construction is well defined. So for every implementation
  of  a  categorial  construction  a  ToDoListEntry  needs to be defined which
  propagates  well  definedness  from  the  input cells to the output. For not
  writing  this  construction  in  every  method, this function can be used to
  install  a  method  which  then installs the correct ToDoListEntries for the
  output.  The  input  syntax  works  exactly  like  InstallMethod,  with  one
  extension:  The  method  creates  an  auxilliary function which computes the
  output  from  the  function  given to InstallMethodWithToDoForIsWellDefined,
  then installs the ToDoListEntries, and then installs the auxilliary function
  instead  of  the  original  one.  This  is normally done with InstallMethod.
  However,  one can change this via the option InstallMethod, which can be set
  to  any  other function which is then used instead of InstallMethod. This is
  used for the caching functions.
  
  10.2-2 InstallSetWithToDoForIsWellDefined
  
  InstallSetWithToDoForIsWellDefined( arg1, arg2, arg3 )  operation
  
  For  the  caching one needs the possibility to install setters for functions
  with  multiple arguments. This is a setter function which automatically adds
  ToDoListEntries  for  IsWellDefined  like  described  above for the manually
  setted result of a method.
  
  10.2-3 DeclareAttributeWithToDoForIsWellDefined
  
  DeclareAttributeWithToDoForIsWellDefined( arg )  function
  
  Since  attributes  install  their  setters  themselfes, one needs to declare
  attributes in another way to ensure ToDoListEntries for IsWellDefined in the
  setter  of  an  attribute.  This  function  works like DeclareAttribute, but
  installs  ToDoListEntries  for the setter of the attribute. Please note that
  implementations       still       need       to       be      done      with
  InstallMethodWithToDoForIsWellDefined.
  
  10.2-4 DeclareFamilyProperty
  
  DeclareFamilyProperty( arg )  function
  
  10.2-5 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS
  
  CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS( list, category )  function
  Returns:  Replaced list
  
  The  function  takes  a  list  (of  lists)  of filters or strings, where the
  strings can be category, cell, object, morphism, or twocell. The strings are
  then  recursively replaced by the corresponding filters of the category. The
  replaced list is returned.
  
  10.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS
  
  CAP_INTERNAL_MERGE_FILTER_LISTS( list, additional, list )  function
  Returns:  merged lists
  
  The  first argument should be a dense list with filters, the second a sparse
  list  containing  filters  not longer then the first one. The filters of the
  second  list are then appended (via and) to the filters in the first list at
  the corresponding position, and the resulting list is returned.
  
  10.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT
  
  CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( string, value )  function
  Returns:  option value
  
  Returns the value of the option with name string, or, if this value is fail,
  the object value.
  
  10.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION
  
  CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION( function, symbol_list, loop_multiple )  function
  Returns:  a list of symbols with multiples
  
  The  function  searches  for the appearance of the strings in symbol list on
  the  function  function  and returns a list of pairs, containing the name of
  the  symbol  and  the number of appearance. If the symbol appears in a loop,
  the number of appearance is counted times the loop multiple.
  
  10.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST
  
  CAP_INTERNAL_MERGE_PRECONDITIONS_LIST( list1, list2 )  function
  Returns:  merge list
  
  The  function  takes  two  lists  containing  pairs of symbols (strings) and
  multiples.  The lists are merged that pairs where the string only appears in
  one  list  is then added to the return list, if a pair with a string appears
  in  both  lists,  the resulting lists only contains this pair once, with the
  higher multiple from both lists.