  
  [1X10 [33X[0;0YTechnical Details[133X[101X
  
  
  [1X10.1 [33X[0;0YThe Category Cat[133X[101X
  
  [1X10.1-1 ObjectCache[101X
  
  [29X[2XObjectCache[102X( [3Xfunctor[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YIsCachingObject[133X
  
  [33X[0;0YRetuns  the  caching  object which stores the results of the functor [3Xfunctor[103X
  applied to objects.[133X
  
  [1X10.1-2 MorphismCache[101X
  
  [29X[2XMorphismCache[102X( [3Xfunctor[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YIsCachingObject[133X
  
  [33X[0;0YRetuns  the  caching  object which stores the results of the functor [3Xfunctor[103X
  applied to morphisms.[133X
  
  
  [1X10.2 [33X[0;0YInstall Functions for IsWellDefined[133X[101X
  
  [1X10.2-1 InstallMethodWithToDoForIsWellDefined[101X
  
  [29X[2XInstallMethodWithToDoForIsWellDefined[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0YThe  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.[133X
  
  [1X10.2-2 InstallSetWithToDoForIsWellDefined[101X
  
  [29X[2XInstallSetWithToDoForIsWellDefined[102X( [3Xarg1[103X, [3Xarg2[103X, [3Xarg3[103X ) [32X operation
  
  [33X[0;0YFor  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.[133X
  
  [1X10.2-3 DeclareAttributeWithToDoForIsWellDefined[101X
  
  [29X[2XDeclareAttributeWithToDoForIsWellDefined[102X( [3Xarg[103X ) [32X function
  
  [33X[0;0YSince  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.[133X
  
  [1X10.2-4 DeclareFamilyProperty[101X
  
  [29X[2XDeclareFamilyProperty[102X( [3Xarg[103X ) [32X function
  
  [1X10.2-5 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS[101X
  
  [29X[2XCAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS[102X( [3Xlist[103X, [3Xcategory[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YReplaced list[133X
  
  [33X[0;0YThe  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.[133X
  
  [1X10.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS[101X
  
  [29X[2XCAP_INTERNAL_MERGE_FILTER_LISTS[102X( [3Xlist[103X, [3Xadditional[103X, [3Xlist[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ymerged lists[133X
  
  [33X[0;0YThe  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.[133X
  
  [1X10.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT[101X
  
  [29X[2XCAP_INTERNAL_RETURN_OPTION_OR_DEFAULT[102X( [3Xstring[103X, [3Xvalue[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Yoption value[133X
  
  [33X[0;0YReturns the value of the option with name string, or, if this value is fail,
  the object value.[133X
  
  [1X10.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION[101X
  
  [29X[2XCAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION[102X( [3Xfunction[103X, [3Xsymbol_list[103X, [3Xloop_multiple[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ya list of symbols with multiples[133X
  
  [33X[0;0YThe  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.[133X
  
  [1X10.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST[101X
  
  [29X[2XCAP_INTERNAL_MERGE_PRECONDITIONS_LIST[102X( [3Xlist1[103X, [3Xlist2[103X ) [32X function
  [6XReturns:[106X  [33X[0;10Ymerge list[133X
  
  [33X[0;0YThe  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.[133X
  
  
  [1X10.3 [33X[0;0YUniversal Objects[133X[101X
  
  [1X10.3-1 WasCreatedAsKernelObject[101X
  
  [29X[2XWasCreatedAsKernelObject[102X[32X filter
  
  [33X[0;0YWhen  created,  this filter is set to true for a kernel object. Note that we
  chose  [10XWasCreatedAsKernelObject[110X  to  be  a  filter  rather  than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-2 WasCreatedAsCokernelObject[101X
  
  [29X[2XWasCreatedAsCokernelObject[102X[32X filter
  
  [33X[0;0YWhen created, this filter is set to true for a cokernel object. Note that we
  chose  [10XWasCreatedAsCokernelObject[110X  to  be  a  filter rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-3 WasCreatedAsZeroObject[101X
  
  [29X[2XWasCreatedAsZeroObject[102X[32X filter
  
  [33X[0;0YWhen  created,  this  filter  is set to true for a zero object. Note that we
  chose  [10XWasCreatedAsZeroObject[110X to be a filter rather than a property, because
  by default, a filter is set to false.[133X
  
  [1X10.3-4 WasCreatedAsTerminalObject[101X
  
  [29X[2XWasCreatedAsTerminalObject[102X[32X filter
  
  [33X[0;0YWhen created, this filter is set to true for a terminal object. Note that we
  chose  [10XWasCreatedAsTerminalObject[110X  to  be  a  filter rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-5 WasCreatedAsInitialObject[101X
  
  [29X[2XWasCreatedAsInitialObject[102X[32X filter
  
  [33X[0;0YWhen created, this filter is set to true for an initial object. Note that we
  chose  [10XWasCreatedAsInitialObject[110X  to  be  a  filter  rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-6 WasCreatedAsDirectSum[101X
  
  [29X[2XWasCreatedAsDirectSum[102X[32X filter
  
  [33X[0;0YWhen  created, this filter is set to true for a direct sum object. Note that
  we  chose  [10XWasCreatedAsDirectSum[110X  to  be  a  filter  rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-7 WasCreatedAsCoproduct[101X
  
  [29X[2XWasCreatedAsCoproduct[102X[32X filter
  
  [33X[0;0YWhen  created,  this filter is set to true for a coproduct object. Note that
  we  chose  [10XWasCreatedAsCoprodcut[110X  to  be  a  filter  rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-8 WasCreatedAsDirectProduct[101X
  
  [29X[2XWasCreatedAsDirectProduct[102X[32X filter
  
  [33X[0;0YWhen created, this filter is set to true for a terminal object. Note that we
  chose  [10XWasCreatedAsDirectProduct[110X  to  be  a  filter  rather than a property,
  because by default, a filter is set to false.[133X
  
  [1X10.3-9 WasCreatedAsFiberProduct[101X
  
  [29X[2XWasCreatedAsFiberProduct[102X[32X filter
  
  [33X[0;0YWhen  created, this filter is set to true for a pullback. Note that we chose
  [10XWasCreatedAsFiberProduct[110X  to  be a filter rather than a property, because by
  default, a filter is set to false.[133X
  
  [1X10.3-10 WasCreatedAsPushout[101X
  
  [29X[2XWasCreatedAsPushout[102X[32X filter
  
  [33X[0;0YWhen  created,  this filter is set to true for a pushout. Note that we chose
  [10XWasCreatedAsPushout[110X  to  be  a  filter  rather  than  a property, because by
  default, a filter is set to false.[133X
  
  [1X10.3-11 WasCreatedAsImageObject[101X
  
  [29X[2XWasCreatedAsImageObject[102X[32X filter
  
  [33X[0;0YWhen  created,  this  filter is set to true for an image. Note that we chose
  [10XWasCreatedAsImageObject[110X  to  be  a filter rather than a property, because by
  default, a filter is set to false.[133X
  
