  
  [1X4 [33X[0;0YObjects[133X[101X
  
  [33X[0;0YAny  GAP  object which is IsCapCategoryObject can be added to a category and
  then  becomes an object in this category. Any object can belong to one or no
  category. After a GAP object is added to the category, it knows which things
  can  be  computed in its category and to which category it belongs. It knows
  categorial  properties  and  attributes,  and  the functions for existential
  quantifiers can be applied to the object.[133X
  
  
  [1X4.1 [33X[0;0YAttributes for the Type of Objects[133X[101X
  
  [1X4.1-1 CapCategory[101X
  
  [29X[2XCapCategory[102X( [3Xa[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument is an object [23Xa[123X. The output is the category [23X\mathbf{C}[123X to which
  [23Xa[123X was added.[133X
  
  
  [1X4.2 [33X[0;0YEquality for Objects[133X[101X
  
  [1X4.2-1 IsEqualForObjects[101X
  
  [29X[2XIsEqualForObjects[102X( [3Xa[103X, [3Xb[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa[123X  and  [23Xb[123X.  The  output is [10Xtrue[110X if [23Xa = b[123X,
  otherwise the output is [10Xfalse[110X.[133X
  
  [1X4.2-2 AddIsEqualForObjects[101X
  
  [29X[2XAddIsEqualForObjects[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function [23XF[123X to the category for the basic operation [10XIsEqualForObjects[110X.
  [23XF: (a,b) \mapsto \mathtt{IsEqualForObjects}(a,b)[123X.[133X
  
  
  [1X4.3 [33X[0;0YCategorical Properties of Objects[133X[101X
  
  [1X4.3-1 AddIsProjective[101X
  
  [29X[2XAddIsProjective[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function [23XF[123X to the category for the basic operation [10XIsProjective[110X. [23XF: a
  \mapsto \mathtt{IsProjective}(a)[123X.[133X
  
  [1X4.3-2 AddIsInjective[101X
  
  [29X[2XAddIsInjective[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function  [23XF[123X to the category for the basic operation [10XIsInjective[110X. [23XF: a
  \mapsto \mathtt{IsInjective}(a)[123X.[133X
  
  [1X4.3-3 AddIsTerminal[101X
  
  [29X[2XAddIsTerminal[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function  [23XF[123X  to the category for the basic operation [10XIsTerminal[110X. [23XF: a
  \mapsto \mathtt{IsTerminal}(a)[123X.[133X
  
  [1X4.3-4 AddIsInitial[101X
  
  [29X[2XAddIsInitial[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function  [23XF[123X  to  the category for the basic operation [10XIsInitial[110X. [23XF: a
  \mapsto \mathtt{IsInitial}(a)[123X.[133X
  
  [1X4.3-5 IsZeroForObjects[101X
  
  [29X[2XIsZeroForObjects[102X( [3Xa[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument is an object [23Xa[123X of a category [23X\mathbf{C}[123X. The output is [10Xtrue[110X if
  [23Xa[123X  is  isomorphic  to the zero object of [23X\mathbf{C}[123X, otherwise the output is
  [10Xfalse[110X.[133X
  
  [1X4.3-6 AddIsZeroForObjects[101X
  
  [29X[2XAddIsZeroForObjects[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function  [23XF[123X to the category for the basic operation [10XIsZeroForObjects[110X.
  [23XF: a \mapsto \mathtt{IsZeroForObjects}(a)[123X.[133X
  
  
  [1X4.4 [33X[0;0YTool functions for caches[133X[101X
  
  [1X4.4-1 IsEqualForCacheForObjects[101X
  
  [29X[2XIsEqualForCacheForObjects[102X( [3Xphi[103X, [3Xpsi[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YCompares two objects in the cache[133X
  
  [1X4.4-2 AddIsEqualForCacheForObjects[101X
  
  [29X[2XAddIsEqualForCacheForObjects[102X( [3Xc[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynorthing[133X
  
  [33X[0;0YBy  default,  CAP uses caches to store the values of Categorical operations.
  To  get  a value out of the cache, one needs to compare the input of a basic
  operation  with  its  previous  input.  To  compare objects in the category,
  IsEqualForCacheForObject   is   used.  By  default  this  is  an  alias  for
  IsEqualForObjects,  where  fail  is  substituted  by  false.  If  you  add a
  function,  this  function  used  instead.  A function [23XF: a,b \mapsto bool[123X is
  expected  here.  The  output has to be true or false. Fail is not allowed in
  this context.[133X
  
  
  [1X4.5 [33X[0;0YAdding Objects to a Category[133X[101X
  
  [1X4.5-1 AddObjectRepresentation[101X
  
  [29X[2XAddObjectRepresentation[102X( [3Xcategory[103X, [3Xfilter[103X ) [32X operation
  
  [33X[0;0YThe  argument  [3Xfilter[103X  is  used  to  create  an object type for the category
  [3Xcategory[103X,  which  is  then  used  in  [10XObjectifyObjectForCAPWithAttributes[110X to
  objectify objects for this category.[133X
  
  [1X4.5-2 ObjectifyObjectForCAPWithAttributes[101X
  
  [29X[2XObjectifyObjectForCAPWithAttributes[102X( [3Xobject[103X, [3Xcategory[103X[, [3Xattribute1[103X, [3Xvalue1[103X, [3X...[103X] ) [32X function
  
  [33X[0;0YObjectifies  the  object  [3Xobject[103X  with  the  type created for objects in the
  category  [3Xcategory[103X.  The  type  is  created  by  passing a representation to
  [10XAddObjectRepresentation[110X.  Objects which are objectified using this method do
  not have to be passed to the [10XAddObject[110X function.[133X
  
  
  [1X4.6 [33X[0;0YWell-Definedness of Objects[133X[101X
  
  [1X4.6-1 IsWellDefinedForObjects[101X
  
  [29X[2XIsWellDefinedForObjects[102X( [3Xa[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output is [10Xtrue[110X if [23Xa[123X is well-defined,
  otherwise the output is [10Xfalse[110X.[133X
  
  [1X4.6-2 AddIsWellDefinedForObjects[101X
  
  [29X[2XAddIsWellDefinedForObjects[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsWellDefinedForObjects[110X. [23XF: a \mapsto \mathtt{IsWellDefinedForObjects}( a )[123X.[133X
  
  
  [1X4.7 [33X[0;0YProjectives[133X[101X
  
  [33X[0;0YFor  a  given object [23XA[123X in an abelian category having enough projectives, the
  following  commands  allow  us  to compute some projective object [23XP[123X together
  with an epimorphism [23X\pi: P \rightarrow A[123X.[133X
  
  [1X4.7-1 SomeProjectiveObject[101X
  
  [29X[2XSomeProjectiveObject[102X( [3XA[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  argument  is  an  object  [23XA[123X. The output is some projective object [23XP[123X for
  which there exists an epimorphism [23X\pi: P \rightarrow A[123X.[133X
  
  [1X4.7-2 EpimorphismFromSomeProjectiveObject[101X
  
  [29X[2XEpimorphismFromSomeProjectiveObject[102X( [3XA[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(P,A)[123X[133X
  
  [33X[0;0YThe argument is an object [23XA[123X. The output is an epimorphism [23X\pi: P \rightarrow
  A[123X    with    [23XP[123X   a   projective   object   that   equals   the   output   of
  [23X\mathrm{SomeProjectiveObject}(A)[123X.[133X
  
  [1X4.7-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject[101X
  
  [29X[2XEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject[102X( [3XA[103X, [3XP[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(P,A)[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23XA[123X and a projective object [23XP[123X that equals the
  output  of  [23X\mathrm{SomeProjectiveObject}(A)[123X.  The  output is an epimorphism
  [23X\pi: P \rightarrow A[123X.[133X
  
  [1X4.7-4 ProjectiveLift[101X
  
  [29X[2XProjectiveLift[102X( [3Xpi[103X, [3Xepsilon[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(P,B)[123X[133X
  
  [33X[0;0YThe  arguments  are a morphism [23X\pi: P \rightarrow A[123X with [23XP[123X a projective, and
  an  epimorphism [23X\epsilon: B \rightarrow A[123X. The output is a morphism [23X\lambda:
  P \rightarrow B[123X such that [23X\epsilon \circ \lambda = \pi[123X.[133X
  
  [1X4.7-5 AddSomeProjectiveObject[101X
  
  [29X[2XAddSomeProjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XSomeProjectiveObject[110X. [23XF: A \mapsto P[123X.[133X
  
  [1X4.7-6 AddEpimorphismFromSomeProjectiveObject[101X
  
  [29X[2XAddEpimorphismFromSomeProjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XEpimorphismFromSomeProjectiveObject[110X. [23XF: A \mapsto \pi[123X.[133X
  
  [1X4.7-7 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject[101X
  
  [29X[2XAddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject[110X.      [23XF:
  (A,P) \mapsto \pi[123X.[133X
  
  [1X4.7-8 AddProjectiveLift[101X
  
  [29X[2XAddProjectiveLift[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given function [23XF[123X to the category for the basic operation [10XProjectiveLift[110X. The
  function [23XF[123X maps a pair [23X(\pi, \epsilon)[123X to a projective lift [23X\lambda[123X.[133X
  
  
  [1X4.8 [33X[0;0YInjectives[133X[101X
  
  [33X[0;0YFor  a  given  object [23XA[123X in an abelian category having enough injectives, the
  following commands allow us to compute some injective object [23XI[123X together with
  a monomorphism [23X\iota: A \rightarrow I[123X.[133X
  
  [1X4.8-1 SomeInjectiveObject[101X
  
  [29X[2XSomeInjectiveObject[102X( [3XA[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe argument is an object [23XA[123X. The output is some injective object [23XI[123X for which
  there exists a monomorphism [23X\iota: A \rightarrow I[123X.[133X
  
  [1X4.8-2 MonomorphismIntoSomeInjectiveObject[101X
  
  [29X[2XMonomorphismIntoSomeInjectiveObject[102X( [3XA[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(I,A)[123X[133X
  
  [33X[0;0YThe  argument  is  an  object  [23XA[123X.  The  output  is  a  monomorphism [23X\iota: A
  \rightarrow  I[123X  with  [23XI[123X  an  injective  object  that  equals  the  output of
  [23X\mathrm{SomeInjectiveObject}(A)[123X.[133X
  
  [1X4.8-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject[101X
  
  [29X[2XMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject[102X( [3XA[103X, [3XI[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(I,A)[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23XA[123X and an injective object [23XI[123X that equals the
  output  of  [23X\mathrm{SomeInjectiveObject}(A)[123X.  The  output  is a monomorphism
  [23X\iota: A \rightarrow I[123X.[133X
  
  [1X4.8-4 InjectiveColift[101X
  
  [29X[2XInjectiveColift[102X( [3X\iota[103X, [3X\beta[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,I)[123X[133X
  
  [33X[0;0YThe arguments are a morphism [23X\iota: B \rightarrow A[123X and [23X\beta: B \rightarrow
  I[123X  where  [23XI[123X  is  an  injective  object.  The output is a morphism [23X\lambda: A
  \rightarrow I[123X such that [23X\lambda \circ \iota = \beta[123X.[133X
  
  [1X4.8-5 AddSomeInjectiveObject[101X
  
  [29X[2XAddSomeInjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XSomeInjectiveObject[110X. [23XF: A \mapsto I[123X.[133X
  
  [1X4.8-6 AddMonomorphismIntoSomeInjectiveObject[101X
  
  [29X[2XAddMonomorphismIntoSomeInjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonomorphismIntoSomeInjectiveObject[110X. [23XF: A \mapsto \pi[123X.[133X
  
  [1X4.8-7 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject[101X
  
  [29X[2XAddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject[110X. [23XF: (A,I)
  \mapsto \pi[123X.[133X
  
  [1X4.8-8 AddInjectiveColift[101X
  
  [29X[2XAddInjectiveColift[102X( [3XC[103X, [3XF[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a category [23XC[123X and a function [23XF[123X. This operations adds the
  given  function  [23XF[123X  to the category for the basic operation [10XInjectiveColift[110X.
  The  function [23XF[123X maps a pair [23X(\iota, \beta)[123X to an injective colift [23X\lambda[123X if
  it exists, and to [10Xfail[110X otherwise.[133X
  
