  
  [1X3 [33X[0;0YMorphisms[133X[101X
  
  [33X[0;0YAny  GAP  object satisfying [10XIsCapCategoryMorphism[110X can be added to a category
  and then becomes a morphism in this category. Any morphism 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   categorical   properties  and  attributes,  and  the  functions  for
  existential quantifiers can be applied to the morphism.[133X
  
  
  [1X3.1 [33X[0;0YAttributes for the Type of Morphisms[133X[101X
  
  [1X3.1-1 CapCategory[101X
  
  [33X[1;0Y[29X[2XCapCategory[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument is a morphism [23X\alpha[123X. The output is the category [23X\mathbf{C}[123X to
  which [23X\alpha[123X was added.[133X
  
  [1X3.1-2 Source[101X
  
  [33X[1;0Y[29X[2XSource[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe argument is a morphism [23X\alpha: a \rightarrow b[123X. The output is its source
  [23Xa[123X.[133X
  
  [1X3.1-3 Range[101X
  
  [33X[1;0Y[29X[2XRange[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  argument is a morphism [23X\alpha: a \rightarrow b[123X. The output is its range
  [23Xb[123X.[133X
  
  
  [1X3.2 [33X[0;0YMorphism constructors[133X[101X
  
  [1X3.2-1 MorphismConstructor[101X
  
  [33X[1;0Y[29X[2XMorphismConstructor[102X( [3XS[103X, [3Xa[103X, [3XT[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(S,T)[123X[133X
  
  [33X[0;0YThe  arguments are two objects [23XS[123X and [23XT[123X in a category, and a morphism datum [23Xa[123X
  (type  and  semantics  of  the  morphism  datum depend on the category). The
  output is a morphism in [23X\mathrm{Hom}(S,T)[123X defined by [23Xa[123X. Note that by default
  this  CAP  operation is not cached. You can change this behaviour by calling
  [10XSetCachingToWeak(  C,  "MorphismConstructor"  )[110X  resp. [10XSetCachingToCrisp( C,
  "MorphismConstructor" )[110X.[133X
  
  [1X3.2-2 MorphismDatum[101X
  
  [33X[1;0Y[29X[2XMorphismDatum[102X( [3Xmor[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ydepends on the category[133X
  
  [33X[0;0YThe argument is a CAP category morphism [3Xmor[103X. The output is a datum which can
  be   used   to   construct   [3Xmor[103X,   that   is,   [10XIsEqualForMorphisms(   [110X[3Xmor[103X[10X,
  MorphismConstructor(  Source( [110X[3Xmor[103X[10X ), MorphismDatum( [110X[3Xmor[103X[10X ), Range( [110X[3Xmor[103X[10X ) ) )[110X.
  Note  that  by default this CAP operation is not cached. You can change this
  behaviour   by   calling   [10XSetCachingToWeak(   C,  "MorphismDatum"  )[110X  resp.
  [10XSetCachingToCrisp( C, "MorphismDatum" )[110X.[133X
  
  
  [1X3.3 [33X[0;0YCategorical Properties of Morphisms[133X[101X
  
  [1X3.3-1 IsMonomorphism[101X
  
  [33X[1;0Y[29X[2XIsMonomorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output  is [10Xtrue[110X if [23X\alpha[123X is a
  monomorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-2 IsEpimorphism[101X
  
  [33X[1;0Y[29X[2XIsEpimorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output is [10Xtrue[110X if [23X\alpha[123X is an
  epimorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-3 IsIsomorphism[101X
  
  [33X[1;0Y[29X[2XIsIsomorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output is [10Xtrue[110X if [23X\alpha[123X is an
  isomorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-4 IsSplitMonomorphism[101X
  
  [33X[1;0Y[29X[2XIsSplitMonomorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is a morphism [23X\alpha[123X. The output is [10Xtrue[110X if [23X\alpha[123X is a split
  monomorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-5 IsSplitEpimorphism[101X
  
  [33X[1;0Y[29X[2XIsSplitEpimorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is a morphism [23X\alpha[123X. The output is [10Xtrue[110X if [23X\alpha[123X is a split
  epimorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-6 IsOne[101X
  
  [33X[1;0Y[29X[2XIsOne[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a morphism [23X\alpha: a \rightarrow a[123X. The output is [10Xtrue[110X if
  [23X\alpha[123X is congruent to the identity of [23Xa[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.3-7 IsIdempotent[101X
  
  [33X[1;0Y[29X[2XIsIdempotent[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a morphism [23X\alpha: a \rightarrow a[123X. The output is [10Xtrue[110X if
  [23X\alpha^2 \sim_{a,a} \alpha[123X, otherwise the output is [10Xfalse[110X.[133X
  
  
  [1X3.4 [33X[0;0YRandom Morphisms[133X[101X
  
  [33X[0;0YCAP  provides  two  principal  methods  to generate random morphisms with or
  without fixed source and range:[133X
  
  [30X    [33X[0;6Y[13XBy  integers[113X:  The  integer  is simply a parameter that can be used to
        create a random morphism.[133X
  
  [30X    [33X[0;6Y[13XBy  lists[113X: The list is used when creating a random morphism would need
        more  than  one parameter. Lists offer more flexibility at the expense
        of  the genericity of the methods. This happens because lists that are
        valid as input in some category may be not valid for other categories.
        Hence,  these  operations  are  not  thought  to  be  used  in generic
        categorical algorithms.[133X
  
  [1X3.4-1 RandomMorphismWithFixedSourceByInteger[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceByInteger[102X( [3Xa[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X in a category [23XC[123X and an integer [23Xn[123X. The output
  is a random morphism [23X\alpha: a \rightarrow b[123X for some object [23Xb[123X in [23XC[123X. If [23XC[123X is
  equipped      with      the      methods      [10XRandomObjectByInteger[110X      and
  [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X and [23XC[123X is an Ab-category, then
  [10XRandomMorphismWithFixedSourceByInteger[110X[23X(C,a,n)[123X     can    be    derived    as
  [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X([23XC[123X,[23Xa[123X,[23Xb[123X,[23X1[123X+[10XLog2Int[110X([23Xn[123X))  where  [23Xb[123X
  is computed via [10XRandomObjectByInteger[110X([23XC[123X,[23Xn[123X).[133X
  
  [1X3.4-2 RandomMorphismWithFixedSourceByList[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceByList[102X( [3Xa[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are an object [23Xa[123X in a category [23XC[123X and a list [23XL[123X. The output is a
  random  morphism  [23X\alpha:  a  \rightarrow  b[123X for some object [23Xb[123X in [23XC[123X. If [23XC[123X is
  equipped       with       the       methods      [10XRandomObjectByList[110X      and
  [10XRandomMorphismWithFixedSourceAndRangeByList[110X  and  [23XC[123X  is an Ab-category, then
  [10XRandomMorphismWithFixedSourceByList[110X[23X(C,a,L)[123X     can     be     derived     as
  [10XRandomMorphismWithFixedSourceAndRangeByList[110X([23XC,a,b,L[2][123X)  where [23Xb[123X is computed
  via [10XRandomObjectByList[110X([23XC,L[1][123X).[133X
  
  [1X3.4-3 RandomMorphismWithFixedRangeByInteger[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedRangeByInteger[102X( [3Xb[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xb[123X in a category [23XC[123X and an integer [23Xn[123X. The output
  is a random morphism [23X\alpha: a \rightarrow b[123X for some object [23Xa[123X in [23XC[123X. If [23XC[123X is
  equipped      with      the      methods      [10XRandomObjectByInteger[110X      and
  [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X and [23XC[123X is an Ab-category, then
  [10XRandomMorphismWithFixedRangeByInteger[110X[23X(C,b,n)[123X     can     be    derived    as
  [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X([23XC[123X,[23Xa[123X,[23Xb[123X,[23X1[123X+[10XLog2Int[110X([23Xn[123X))  where  [23Xa[123X
  is computed via [10XRandomObjectByInteger[110X([23XC[123X,[23Xn[123X).[133X
  
  [1X3.4-4 RandomMorphismWithFixedRangeByList[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedRangeByList[102X( [3Xb[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are an object [23Xb[123X in a category [23XC[123X and a list [23XL[123X. The output is a
  random  morphism  [23X\alpha:  a  \rightarrow  b[123X for some object [23Xa[123X in [23XC[123X. If [23XC[123X is
  equipped       with       the       methods      [10XRandomObjectByList[110X      and
  [10XRandomMorphismWithFixedSourceAndRangeByList[110X  and  [23XC[123X  is an Ab-category, then
  [10XRandomMorphismWithFixedRangeByList[110X[23X(C,b,L)[123X      can     be     derived     as
  [10XRandomMorphismWithFixedSourceAndRangeByList[110X([23XC,a,b,L[2][123X)  where [23Xa[123X is computed
  via [10XRandomObjectByList[110X([23XC,L[1][123X).[133X
  
  [1X3.4-5 RandomMorphismWithFixedSourceAndRangeByInteger[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceAndRangeByInteger[102X( [3Xa[103X, [3Xb[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are two objects [23Xa[123X and [23Xb[123X in a category [23XC[123X and an integer [23Xn[123X. The
  output is a random morphism [23X\alpha: a \rightarrow b[123X in [23XC[123X.[133X
  
  [1X3.4-6 RandomMorphismWithFixedSourceAndRangeByList[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceAndRangeByList[102X( [3Xa[103X, [3Xb[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThis  operation is not a CAP basic operation The arguments are two objects [23Xa[123X
  and  [23Xb[123X in a category [23XC[123X and a list [23XL[123X. The output is a random morphism [23X\alpha:
  a \rightarrow b[123X in [23XC[123X.[133X
  
  [1X3.4-7 RandomMorphismByInteger[101X
  
  [33X[1;0Y[29X[2XRandomMorphismByInteger[102X( [3XC[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23XC[123X[133X
  
  [33X[0;0YThe  arguments  are  a  category  [23XC[123X and an integer [23Xn[123X. The output is a random
  morphism in [23XC[123X. The operation can be derived in three different ways:[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped  with  the  methods  [10XRandomObjectByInteger[110X  and
        [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X    and    [23XC[123X    is    an
        Ab-category,    then    [10XRandomMorphism[110X[23X(C,n)[123X    can   be   derived   as
        [10XRandomMorphismWithFixedSourceAndRangeByInteger[110X([23XC,a,b[123X,[23X1[123X+[10XLog2Int[110X([23Xn[123X))
        where [23Xa[123X and [23Xb[123X are computed via [10XRandomObjectByInteger[110X([23XC,n[123X).[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped  with  the  methods  [10XRandomObjectByInteger[110X  and
        [10XRandomMorphismWithFixedSourceByInteger[110X,  then  [10XRandomMorphism[110X[23X(C,n)[123X can
        be derived as [10XRandomMorphismWithFixedSourceByInteger[110X([23XC,a,1[123X+[10XLog2Int[110X([23Xn[123X))
        where [23Xa[123X is computed via [10XRandomObjectByInteger[110X([23XC,n[123X).[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped  with  the  methods  [10XRandomObjectByInteger[110X  and
        [10XRandomMorphismWithFixedRangeByInteger[110X, then [10XRandomMorphism[110X[23X(C,n)[123X can be
        derived   as   [10XRandomMorphismWithFixedRangeByInteger[110X([23XC,b,1[123X+[10XLog2Int[110X([23Xn[123X))
        where [23Xb[123X is computed via [10XRandomObjectByInteger[110X([23XC,n[123X).[133X
  
  [1X3.4-8 RandomMorphismByList[101X
  
  [33X[1;0Y[29X[2XRandomMorphismByList[102X( [3XC[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23XC[123X[133X
  
  [33X[0;0YThe arguments are a category [23XC[123X and a list [23XL[123X. The output is a random morphism
  in [23XC[123X. The operation can be derived in three different ways:[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped   with   the   methods  [10XRandomObjectByList[110X  and
        [10XRandomMorphismWithFixedSourceAndRangeByList[110X  and  [23XC[123X is an Ab-category,
        then       [10XRandomMorphism[110X[23X(C,L)[123X       can       be      derived      as
        [10XRandomMorphismWithFixedSourceAndRangeByList[110X([23XC,a,b,L[3][123X)) where [23Xa[123X and [23Xb[123X
        are computed via [10XRandomObjectByList[110X([23XC,L[i][123X) for [23Xi=1,2[123X respectively.[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped   with   the   methods  [10XRandomObjectByList[110X  and
        [10XRandomMorphismWithFixedSourceByList[110X,  then  [10XRandomMorphism[110X[23X(C,L)[123X can be
        derived  as  [10XRandomMorphismWithFixedSourceByList[110X([23XC,a,L[2][123X)  where [23Xa[123X is
        computed via [10XRandomObjectByList[110X([23XC,L[1][123X).[133X
  
  [30X    [33X[0;6YIf   [23XC[123X   is   equipped   with   the   methods  [10XRandomObjectByList[110X  and
        [10XRandomMorphismWithFixedRangeByList[110X,  then  [10XRandomMorphism[110X[23X(C,L)[123X  can be
        derived  as  [10XRandomMorphismWithFixedRangeByList[110X([23XC,b,L[2][123X)  where  [23Xb[123X is
        computed via [10XRandomObjectByList[110X([23XC,L[1][123X).[133X
  
  [1X3.4-9 RandomMorphismWithFixedSource[101X
  
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSource[102X( [3Xa[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSource[102X( [3Xa[103X, [3XL[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphismWithFixedRange[102X( [3Xb[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphismWithFixedRange[102X( [3Xb[103X, [3XL[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceAndRange[102X( [3Xa[103X, [3Xb[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphismWithFixedSourceAndRange[102X( [3Xa[103X, [3Xb[103X, [3XL[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphism[102X( [3Xa[103X, [3Xb[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphism[102X( [3Xa[103X, [3Xb[103X, [3XL[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphism[102X( [3XC[103X, [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomMorphism[102X( [3XC[103X, [3XL[103X ) [32X operation[133X
  
  [33X[0;0YThese  are  convenient methods and they, depending on the input, delegate to
  one of the above methods.[133X
  
  
  [1X3.5 [33X[0;0YNon-Categorical Properties of Morphisms[133X[101X
  
  [33X[0;0YNon-categorical properties are not stable under equivalences of categories.[133X
  
  [1X3.5-1 IsEqualToIdentityMorphism[101X
  
  [33X[1;0Y[29X[2XIsEqualToIdentityMorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a morphism [23X\alpha: a \rightarrow b[123X. The output is [10Xtrue[110X if
  [23X\alpha = \mathrm{id}_a[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.5-2 IsEqualToZeroMorphism[101X
  
  [33X[1;0Y[29X[2XIsEqualToZeroMorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a morphism [23X\alpha: a \rightarrow b[123X. The output is [10Xtrue[110X if
  [23X\alpha = 0[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.5-3 IsEndomorphism[101X
  
  [33X[1;0Y[29X[2XIsEndomorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output is [10Xtrue[110X if [23X\alpha[123X is an
  endomorphism, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.5-4 IsAutomorphism[101X
  
  [33X[1;0Y[29X[2XIsAutomorphism[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output is [10Xtrue[110X if [23X\alpha[123X is an
  automorphism, otherwise the output is [10Xfalse[110X.[133X
  
  
  [1X3.6 [33X[0;0YAdding Morphisms to a Category[133X[101X
  
  [1X3.6-1 Add[101X
  
  [33X[1;0Y[29X[2XAdd[102X( [3Xcategory[103X, [3Xmorphism[103X ) [32X operation[133X
  
  [33X[0;0YAdds [3Xmorphism[103X as a morphism to [3Xcategory[103X.[133X
  
  [1X3.6-2 AddMorphism[101X
  
  [33X[1;0Y[29X[2XAddMorphism[102X( [3Xcategory[103X, [3Xmorphism[103X ) [32X operation[133X
  
  [33X[0;0YAdds  [3Xmorphism[103X  as  a  morphism to [3Xcategory[103X. If [3Xmorphism[103X already lies in the
  filter [10XIsCapCategoryMorphism[110X, the operation [2XAdd[102X ([14X3.6-1[114X) can be used instead.[133X
  
  [1X3.6-3 AddMorphismRepresentation[101X
  
  [33X[1;0Y[29X[2XAddMorphismRepresentation[102X( [3Xcategory[103X, [3Xfilter[103X ) [32X operation[133X
  
  [33X[0;0YThe  argument  [3Xfilter[103X  is  used  to  create a morphism type for the category
  [3Xcategory[103X,          which          is          then          used          in
  [10XObjectifyMorphismWithSourceAndRangeForCAPWithAttributes[110X     to     objectify
  morphisms for this category. [3Xfilter[103X must imply [10XIsCapCategoryMorphism[110X.[133X
  
  [1X3.6-4 ObjectifyMorphismWithSourceAndRangeForCAPWithAttributes[101X
  
  [33X[1;0Y[29X[2XObjectifyMorphismWithSourceAndRangeForCAPWithAttributes[102X( [3Xmorphism[103X, [3Xcategory[103X, [3Xsource[103X, [3Xrange[103X[, [3Xattr1[103X, [3Xval1[103X, [3Xattr2[103X, [3Xval2[103X, [3X...[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YObjectifies the morphism [3Xmorphism[103X with the type created for morphisms in the
  category  [3Xcategory[103X.  The  type  is  created  by  passing a representation to
  [10XAddMorphismRepresentation[110X. Morphisms which are objectified using this method
  do  not  have to be passed to the [10XAddMorphism[110X function. The arguments [10Xsource[110X
  and  [10Xrange[110X are assumed to be objectified. The optional arguments behave like
  the  corresponding  arguments  in  [10XObjectifyWithAttributes[110X. Also returns the
  objectified morphism.[133X
  
  [1X3.6-5 CreateCapCategoryMorphismWithAttributes[101X
  
  [33X[1;0Y[29X[2XCreateCapCategoryMorphismWithAttributes[102X( [3Xmorphism[103X, [3Xcategory[103X, [3Xsource[103X, [3Xrange[103X[, [3Xattr1[103X, [3Xval1[103X, [3Xattr2[103X, [3Xval2[103X, [3X...[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YShorthand  for [10XObjectifyMorphismWithSourceAndRangeForCAPWithAttributes( rec(
  ), category, source, range[, attr1, val1, attr2, val2, ...] )[110X.[133X
  
  
  [1X3.7 [33X[0;0YEquality and Congruence for Morphisms[133X[101X
  
  [1X3.7-1 IsCongruentForMorphisms[101X
  
  [33X[1;0Y[29X[2XIsCongruentForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are two morphisms [23X\alpha, \beta: a \rightarrow b[123X. The output
  is [10Xtrue[110X if [23X\alpha \sim_{a,b} \beta[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.7-2 IsEqualForMorphisms[101X
  
  [33X[1;0Y[29X[2XIsEqualForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are two morphisms [23X\alpha, \beta: a \rightarrow b[123X. The output
  is [10Xtrue[110X if [23X\alpha = \beta[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.7-3 IsEqualForMorphismsOnMor[101X
  
  [33X[1;0Y[29X[2XIsEqualForMorphismsOnMor[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  b,  \beta:  c
  \rightarrow d[123X. The output is [10Xtrue[110X if [23X\alpha = \beta[123X, otherwise the output is
  [10Xfalse[110X.[133X
  
  
  [1X3.8 [33X[0;0YBasic Operations for Morphisms in Ab-Categories[133X[101X
  
  [1X3.8-1 IsZeroForMorphisms[101X
  
  [33X[1;0Y[29X[2XIsZeroForMorphisms[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a morphism [23X\alpha: a \rightarrow b[123X. The output is [10Xtrue[110X if
  [23X\alpha \sim_{a,b} 0[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [1X3.8-2 AdditionForMorphisms[101X
  
  [33X[1;0Y[29X[2XAdditionForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments  are two morphisms [23X\alpha, \beta: a \rightarrow b[123X. The output
  is the addition [23X\alpha + \beta[123X. Note: The addition has to be compatible with
  the congruence of morphisms.[133X
  
  [1X3.8-3 SubtractionForMorphisms[101X
  
  [33X[1;0Y[29X[2XSubtractionForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments  are two morphisms [23X\alpha, \beta: a \rightarrow b[123X. The output
  is the addition [23X\alpha - \beta[123X. Note: The addition has to be compatible with
  the congruence of morphisms.[133X
  
  [1X3.8-4 AdditiveInverseForMorphisms[101X
  
  [33X[1;0Y[29X[2XAdditiveInverseForMorphisms[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  a  \rightarrow b[123X. The output is its
  additive  inverse  [23X-\alpha[123X. Note: The addition has to be compatible with the
  congruence of morphisms.[133X
  
  [1X3.8-5 MultiplyWithElementOfCommutativeRingForMorphisms[101X
  
  [33X[1;0Y[29X[2XMultiplyWithElementOfCommutativeRingForMorphisms[102X( [3Xr[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are an element [23Xr[123X of a commutative ring and a morphism [23X\alpha:
  a  \rightarrow  b[123X.  The output is the multiplication with the ring element [23Xr
  \cdot  \alpha[123X.  Note:  The  multiplication  has  to  be  compatible with the
  congruence of morphisms.[133X
  
  [1X3.8-6 *[101X
  
  [33X[1;0Y[29X[2X*[102X( [3Xr[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThis  is  a  convenience method. It has two arguments. The first argument is
  either  a  rational  number  [23Xq[123X  or an element [23Xr[123X of a commutative ring [23XR[123X. The
  second  argument  is a morphism [23X\alpha: a \rightarrow b[123X in a linear category
  over  the  commutative  ring  [23XR[123X.  In  the  case where the first element is a
  rational  number,  this method tries to interpret [23Xq[123X as an element [23Xr[123X of [23XR[123X via
  [10XR!.interpret_rationals_func[110X.  If  no such interpretation exists, this method
  throws  an  error.  The output is the multiplication with the ring element [23Xr
  \cdot \alpha[123X.[133X
  
  [1X3.8-7 ZeroMorphism[101X
  
  [33X[1;0Y[29X[2XZeroMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are two objects [23Xa[123X and [23Xb[123X. The output is the zero morphism [23X0: a
  \rightarrow b[123X.[133X
  
  
  [1X3.9 [33X[0;0YSubobject and Factorobject Operations[133X[101X
  
  [33X[0;0YSubobjects  of  an  object  [23Xc[123X  are  monomorphisms with range [23Xc[123X and a special
  function  for  comparision.  Similarly,  factorobjects  of  an  object [23Xc[123X are
  epimorphisms with source [23Xc[123X and a special function for comparision.[133X
  
  [1X3.9-1 IsEqualAsSubobjects[101X
  
  [33X[1;0Y[29X[2XIsEqualAsSubobjects[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  two  subobjects  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
  \rightarrow  c[123X.  The  output is [10Xtrue[110X if there exists an isomorphism [23X\iota: a
  \rightarrow  b[123X  such that [23X\beta \circ \iota \sim_{a,c} \alpha[123X, otherwise the
  output is [10Xfalse[110X.[133X
  
  [1X3.9-2 IsEqualAsFactorobjects[101X
  
  [33X[1;0Y[29X[2XIsEqualAsFactorobjects[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  two  factorobjects  [23X\alpha:  c  \rightarrow a[123X, [23X\beta: c
  \rightarrow  b[123X.  The  output is [10Xtrue[110X if there exists an isomorphism [23X\iota: b
  \rightarrow  a[123X  such that [23X\iota \circ \beta \sim_{c,a} \alpha[123X, otherwise the
  output is [10Xfalse[110X.[133X
  
  [1X3.9-3 IsDominating[101X
  
  [33X[1;0Y[29X[2XIsDominating[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YIn  short:  Returns [10Xtrue[110X iff [23X\alpha[123X is smaller than [23X\beta[123X. Full description:
  The  arguments  are  two  subobjects  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
  \rightarrow  c[123X.  The  output  is  [10Xtrue[110X  if  there exists a morphism [23X\iota: a
  \rightarrow  b[123X  such that [23X\beta \circ \iota \sim_{a,c} \alpha[123X, otherwise the
  output is [10Xfalse[110X.[133X
  
  [1X3.9-4 IsCodominating[101X
  
  [33X[1;0Y[29X[2XIsCodominating[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YIn  short:  Returns [10Xtrue[110X iff [23X\alpha[123X is smaller than [23X\beta[123X. Full description:
  The  arguments  are  two  factorobjects  [23X\alpha:  c  \rightarrow a[123X, [23X\beta: c
  \rightarrow  b[123X.  The  output  is  [10Xtrue[110X  if  there exists a morphism [23X\iota: b
  \rightarrow  a[123X  such that [23X\iota \circ \beta \sim_{c,a} \alpha[123X, otherwise the
  output is [10Xfalse[110X.[133X
  
  
  [1X3.10 [33X[0;0YIdentity Morphism and Composition of Morphisms[133X[101X
  
  [1X3.10-1 IdentityMorphism[101X
  
  [33X[1;0Y[29X[2XIdentityMorphism[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,a)[123X[133X
  
  [33X[0;0YThe   argument  is  an  object  [23Xa[123X.  The  output  is  its  identity  morphism
  [23X\mathrm{id}_a[123X.[133X
  
  [1X3.10-2 PreCompose[101X
  
  [33X[1;0Y[29X[2XPreCompose[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a, c )[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  b[123X,  [23X\beta:  b
  \rightarrow  c[123X.  The  output  is  the  composition  [23X\beta  \circ  \alpha:  a
  \rightarrow c[123X.[133X
  
  [1X3.10-3 PreCompose[101X
  
  [33X[1;0Y[29X[2XPreCompose[102X( [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a_1, a_{n+1})[123X[133X
  
  [33X[0;0YThis  is  a  convenience  method.  The argument is a list of morphisms [23XL = (
  \alpha_1:  a_1  \rightarrow  a_2,  \alpha_2:  a_2  \rightarrow  a_3,  \dots,
  \alpha_n:   a_n  \rightarrow  a_{n+1}  )[123X.  The  output  is  the  composition
  [23X\alpha_{n}  \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) )
  )[123X.[133X
  
  [1X3.10-4 PreComposeList[101X
  
  [33X[1;0Y[29X[2XPreComposeList[102X( [3XC[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a_1, a_{n+1})[123X[133X
  
  [33X[0;0YThe  argument  is  a  list of morphisms [23XL = ( \alpha_1: a_1 \rightarrow a_2,
  \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \rightarrow a_{n+1} )[123X in
  [23XC[123X.  The  output  is  the composition [23X\alpha_{n} \circ ( \alpha_{n-1} \circ (
  \dots ( \alpha_2 \circ \alpha_1 ) ) )[123X.[133X
  
  [1X3.10-5 PostCompose[101X
  
  [33X[1;0Y[29X[2XPostCompose[102X( [3Xbeta[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a, c )[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\beta:  b  \rightarrow  c[123X,  [23X\alpha:  a
  \rightarrow  b[123X.  The  output  is  the  composition  [23X\beta  \circ  \alpha:  a
  \rightarrow c[123X.[133X
  
  [1X3.10-6 PostCompose[101X
  
  [33X[1;0Y[29X[2XPostCompose[102X( [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a_1, a_{n+1})[123X[133X
  
  [33X[0;0YThis  is  a  convenience  method.  The argument is a list of morphisms [23XL = (
  \alpha_n:  a_n  \rightarrow  a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n,
  \dots,  \alpha_1:  a_1  \rightarrow  a_2  )[123X.  The  output is the composition
  [23X((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1[123X.[133X
  
  [1X3.10-7 PostComposeList[101X
  
  [33X[1;0Y[29X[2XPostComposeList[102X( [3XC[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a_1, a_{n+1})[123X[133X
  
  [33X[0;0YThe argument is a list of morphisms [23XL = ( \alpha_n: a_n \rightarrow a_{n+1},
  \alpha_{n-1}:  a_{n-1} \rightarrow a_n, \dots, \alpha_1: a_1 \rightarrow a_2
  )[123X.  The  output  is  the  composition [23X((\alpha_{n} \circ \alpha_{n-1}) \circ
  \dots \alpha_2) \circ \alpha_1[123X.[133X
  
  [1X3.10-8 SumOfMorphisms[101X
  
  [33X[1;0Y[29X[2XSumOfMorphisms[102X( [3Xs[103X, [3Xmorphisms[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(s,r)[123X[133X
  
  [33X[0;0YThe  arguments are two objects [3Xs[103X, [3Xr[103X and a list [3Xmorphisms[103X of morphisms from [3Xs[103X
  to  [3Xr[103X.  The  output  is  the  sum  of  all  elements  in  [3Xmorphisms[103X,  or the
  zero-morphism from [3Xs[103X to [3Xr[103X if [3Xmorphisms[103X is empty.[133X
  
  
  [1X3.11 [33X[0;0YWell-Definedness of Morphisms[133X[101X
  
  [1X3.11-1 IsWellDefinedForMorphisms[101X
  
  [33X[1;0Y[29X[2XIsWellDefinedForMorphisms[102X( [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X.  The  output  is  [10Xtrue[110X  if [23X\alpha[123X is
  well-defined, otherwise the output is [10Xfalse[110X.[133X
  
  
  [1X3.12 [33X[0;0YLift/Colift[133X[101X
  
  [30X    [33X[0;6YFor   any  pair  of  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
        \rightarrow  c[123X,  we call each morphism [23X\alpha / \beta: a \rightarrow b[123X
        such  that  [23X\beta  \circ  (\alpha / \beta) \sim_{a,c} \alpha[123X a [13Xlift of
        [23X\alpha[123X along [23X\beta[123X[113X.[133X
  
  [30X    [33X[0;6YFor   any  pair  of  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  a
        \rightarrow  b[123X,  we  call  each  morphism  [23X\alpha  \backslash \beta: c
        \rightarrow  b[123X  such  that  [23X(\alpha  \backslash  \beta)  \circ  \alpha
        \sim_{a,b} \beta[123X a [13X colift of [23X\beta[123X along [23X\alpha[123X[113X.[133X
  
  [33X[0;0YNote  that  such lifts (or colifts) do not have to be unique. So in general,
  we  do  not expect that algorithms computing lifts (or colifts) do this in a
  functorial  way.  Thus  the operations [23X\mathtt{Lift}[123X and [23X\mathtt{Colift}[123X are
  not   regarded   as   categorical  operations,  but  only  as  set-theoretic
  operations.[133X
  
  [1X3.12-1 LiftAlongMonomorphism[101X
  
  [33X[1;0Y[29X[2XLiftAlongMonomorphism[102X( [3Xiota[103X, [3Xtau[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(t,k)[123X[133X
  
  [33X[0;0YThe  arguments  are a monomorphism [23X\iota: k \hookrightarrow a[123X and a morphism
  [23X\tau:  t \rightarrow a[123X such that there is a morphism [23Xu: t \rightarrow k[123X with
  [23X\iota \circ u \sim_{t,a} \tau[123X. The output is such a [23Xu[123X.[133X
  
  [1X3.12-2 ColiftAlongEpimorphism[101X
  
  [33X[1;0Y[29X[2XColiftAlongEpimorphism[102X( [3Xepsilon[103X, [3Xtau[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(c,t)[123X[133X
  
  [33X[0;0YThe  arguments  are  an epimorphism [23X\epsilon: a \rightarrow c[123X and a morphism
  [23X\tau:  a \rightarrow t[123X such that there is a morphism [23Xu: c \rightarrow t[123X with
  [23Xu \circ \epsilon \sim_{a,t} \tau[123X. The output is such a [23Xu[123X.[133X
  
  [1X3.12-3 IsLiftableAlongMonomorphism[101X
  
  [33X[1;0Y[29X[2XIsLiftableAlongMonomorphism[102X( [3Xiota[103X, [3Xtau[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are a monomorphism [23X\iota: k \hookrightarrow a[123X and a morphism
  [23X\tau:  t  \rightarrow  a[123X. The output is [10Xtrue[110X if there exists a morphism [23Xu: t
  \rightarrow  k[123X  with [23X\iota \circ u \sim_{t,a} \tau[123X. Otherwise, the output is
  [10Xfalse[110X.[133X
  
  [1X3.12-4 IsColiftableAlongEpimorphism[101X
  
  [33X[1;0Y[29X[2XIsColiftableAlongEpimorphism[102X( [3Xepsilon[103X, [3Xtau[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  an epimorphism [23X\epsilon: a \rightarrow c[123X and a morphism
  [23X\tau:  a  \rightarrow  t[123X. The output is [10Xtrue[110X if there exists a morphism [23Xu: c
  \rightarrow  t[123X  with [23Xu \circ \epsilon \sim_{a,t} \tau[123X. Otherwise, the output
  is [10Xfalse[110X.[133X
  
  [1X3.12-5 Lift[101X
  
  [33X[1;0Y[29X[2XLift[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
  \rightarrow  c[123X  such  that  a lift [23X\alpha / \beta: a \rightarrow b[123X of [23X\alpha[123X
  along  [23X\beta[123X exists. The output is such a lift [23X\alpha / \beta: a \rightarrow
  b[123X.  Recall that a lift [23X\alpha / \beta: a \rightarrow b[123X of [23X\alpha[123X along [23X\beta[123X
  is a morphism such that [23X\beta \circ (\alpha / \beta) \sim_{a,c} \alpha[123X.[133X
  
  [1X3.12-6 LiftOrFail[101X
  
  [33X[1;0Y[29X[2XLiftOrFail[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b) + \{ \mathtt{fail} \}[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
  \rightarrow  c[123X.  The  output  is  a  lift [23X\alpha / \beta: a \rightarrow b[123X of
  [23X\alpha[123X  along  [23X\beta[123X  if  such a lift exists or [23X\mathtt{fail}[123X if it doesn't.
  Recall  that a lift [23X\alpha / \beta: a \rightarrow b[123X of [23X\alpha[123X along [23X\beta[123X is
  a morphism such that [23X\beta \circ (\alpha / \beta) \sim_{a,c} \alpha[123X.[133X
  
  [1X3.12-7 IsLiftable[101X
  
  [33X[1;0Y[29X[2XIsLiftable[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  b
  \rightarrow  c[123X.  The output is [10Xtrue[110X if there exists a lift [23X\alpha / \beta: a
  \rightarrow  b[123X of [23X\alpha[123X along [23X\beta[123X, i.e., a morphism such that [23X\beta \circ
  (\alpha / \beta) \sim_{a,c} \alpha[123X. Otherwise, the output is [10Xfalse[110X.[133X
  
  [1X3.12-8 Colift[101X
  
  [33X[1;0Y[29X[2XColift[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(c,b)[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  a
  \rightarrow b[123X such that a colift [23X\alpha \backslash \beta: c \rightarrow b[123X of
  [23X\beta[123X  along  [23X\alpha[123X  exists.  The output is such a colift [23X\alpha \backslash
  \beta:  c  \rightarrow  b[123X.  Recall  that a colift [23X\alpha \backslash \beta: c
  \rightarrow  b[123X  of  [23X\beta[123X  along  [23X\alpha[123X  is  a  morphism  such that [23X(\alpha
  \backslash \beta) \circ \alpha \sim_{a,b} \beta[123X.[133X
  
  [1X3.12-9 ColiftOrFail[101X
  
  [33X[1;0Y[29X[2XColiftOrFail[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(c,b) + \{ \mathtt{fail} \}[123X[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  a
  \rightarrow b[123X. The output is a colift [23X\alpha \backslash \beta: c \rightarrow
  b[123X  of  [23X\beta[123X  along  [23X\alpha[123X  if  such a colift exists or [23X\mathtt{fail}[123X if it
  doesn't.  Recall  that  a colift [23X\alpha \backslash \beta: c \rightarrow b[123X of
  [23X\beta[123X  along  [23X\alpha[123X is a morphism such that [23X(\alpha \backslash \beta) \circ
  \alpha \sim_{a,b} \beta[123X.[133X
  
  [1X3.12-10 IsColiftable[101X
  
  [33X[1;0Y[29X[2XIsColiftable[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe   arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  c[123X,  [23X\beta:  a
  \rightarrow b[123X. The output is [10Xtrue[110X if there exists a colift [23X\alpha \backslash
  \beta:  c  \rightarrow  b[123X of [23X\beta[123X along [23X\alpha[123X., i.e., a morphism such that
  [23X(\alpha  \backslash  \beta)  \circ  \alpha  \sim_{a,b} \beta[123X. Otherwise, the
  output is [10Xfalse[110X.[133X
  
  
  [1X3.13 [33X[0;0YInverses[133X[101X
  
  [33X[0;0YLet  [23X\alpha:  a  \rightarrow  b[123X  be  a  morphism.  An inverse of [23X\alpha[123X is a
  morphism  [23X\alpha^{-1}:  b  \rightarrow  a[123X such that [23X\alpha \circ \alpha^{-1}
  \sim_{b,b}   \mathrm{id}_b[123X   and   [23X\alpha^{-1}   \circ   \alpha   \sim_{a,a}
  \mathrm{id}_a[123X.[133X
  
  [1X3.13-1 InverseForMorphisms[101X
  
  [33X[1;0Y[29X[2XInverseForMorphisms[102X( [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(b,a)[123X[133X
  
  [33X[0;0YThe  argument  is  an isomorphism [23X\alpha: a \rightarrow b[123X. The output is its
  inverse [23X\alpha^{-1}: b \rightarrow a[123X.[133X
  
  [1X3.13-2 PreInverseForMorphisms[101X
  
  [33X[1;0Y[29X[2XPreInverseForMorphisms[102X( [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(b,a)[123X[133X
  
  [33X[0;0YThe argument is a split-epimorphism [23X\alpha: a \rightarrow b[123X. The output is a
  pre-inverse  [23X\iota:  b \rightarrow a[123X of [23X\alpha[123X, i.e., [23X\iota[123X satisfies [23X\alpha
  \circ  \iota \sim_{b,b} \mathrm{id}_b[123X. The morphism [23X\iota[123X is also known as a
  section or a right-inverse of [23X\alpha[123X.[133X
  
  [1X3.13-3 PostInverseForMorphisms[101X
  
  [33X[1;0Y[29X[2XPostInverseForMorphisms[102X( [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(b,a)[123X[133X
  
  [33X[0;0YThe  argument is a split-monomorphism [23X\alpha: a \rightarrow b[123X. The output is
  a post-inverse [23X\pi: b \rightarrow a[123X of [23X\alpha[123X, i.e., [23X\pi[123X satisfies [23X\pi \circ
  \alpha  \sim_{a,a}  \mathrm{id}_a[123X.  The  morphism  [23X\pi[123X  is  also  known as a
  contraction or a left-inverse of [23X\alpha[123X.[133X
  
  
  [1X3.14 [33X[0;0YTool functions for caches[133X[101X
  
  [1X3.14-1 IsEqualForCacheForMorphisms[101X
  
  [33X[1;0Y[29X[2XIsEqualForCacheForMorphisms[102X( [3Xphi[103X, [3Xpsi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[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 morphisms in the category,
  IsEqualForCacheForMorphisms is used. By default, IsEqualForCacheForMorphisms
  falls  back  to IsEqualForCache (see ToolsForHomalg), which in turn defaults
  to  recursive comparison for lists and [10XIsIdenticalObj[110X in all other cases. If
  you add a function via [10XAddIsEqualForCacheForMorphisms[110X, that function is used
  instead. A function [23XF: a,b \mapsto bool[123X is expected there. The output has to
  be true or false. Fail is not allowed in this context.[133X
  
  
  [1X3.15 [33X[0;0YIsHomSetInhabited[133X[101X
  
  [1X3.15-1 IsHomSetInhabited[101X
  
  [33X[1;0Y[29X[2XIsHomSetInhabited[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments are two objects [3XA[103X and [3XB[103X. The output is [10Xtrue[110X if there exists a
  morphism from [3XA[103X to [3XB[103X, otherwise the output is [10Xfalse[110X.[133X
  
  
  [1X3.16 [33X[0;0YHomomorphism structures[133X[101X
  
  [33X[0;0YHomomorphism  structures  are way to "oversee" the homomorphisms between two
  given  objects.  Let  [23XC[123X,  [23XD[123X  be categories. A [23XD[123X-homomorphism structure for [23XC[123X
  consists of the following data:[133X
  
  [30X    [33X[0;6Ya  functor [23XH: C^{\mathrm{op}} \times C \rightarrow D[123X (when [23XC[123X and [23XD[123X are
        Ab-categories, [23XH[123X is assumed to be bilinear).[133X
  
  [30X    [33X[0;6Yan object [23X1 \in D[123X, called the distinguished object,[133X
  
  [30X    [33X[0;6Ya  bijection  [23X\nu:  \mathrm{Hom}_{C}(a,b)  \simeq  \mathrm{Hom}_{D}(1,
        H(a,b))[123X natural in [23Xa,b \in C[123X.[133X
  
  [1X3.16-1 HomomorphismStructureOnObjects[101X
  
  [33X[1;0Y[29X[2XHomomorphismStructureOnObjects[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object in [23XD[123X[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa, b[123X in [23XC[123X. The output is the value of the
  homomorphism structure on objects [23XH(a,b)[123X.[133X
  
  [1X3.16-2 HomomorphismStructureOnMorphisms[101X
  
  [33X[1;0Y[29X[2XHomomorphismStructureOnMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(H(a',b), H(a,b'))[123X[133X
  
  [33X[0;0YThe  arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  a',  \beta:  b
  \rightarrow  b'[123X  in [23XC[123X. The output is the value of the homomorphism structure
  on morphisms [23XH(\alpha, \beta )[123X.[133X
  
  [1X3.16-3 HomomorphismStructureOnMorphismsWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XHomomorphismStructureOnMorphismsWithGivenObjects[102X( [3Xs[103X, [3Xalpha[103X, [3Xbeta[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(H(a',b), H(a,b'))[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  =  H(a',b)[123X in [23XD[123X, two morphisms [23X\alpha: a
  \rightarrow  a',  \beta: b \rightarrow b'[123X in [23XC[123X, and an object [23Xr = H(a,b')[123X in
  [23XD[123X.  The  output  is  the  value  of  the homomorphism structure on morphisms
  [23XH(\alpha, \beta )[123X.[133X
  
  [1X3.16-4 DistinguishedObjectOfHomomorphismStructure[101X
  
  [33X[1;0Y[29X[2XDistinguishedObjectOfHomomorphismStructure[102X( [3XC[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object in [23XD[123X[133X
  
  [33X[0;0YThe  argument is a category [23XC[123X. The output is the distinguished object [23X1[123X in [23XD[123X
  of the homomorphism structure.[133X
  
  [1X3.16-5 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure[101X
  
  [33X[1;0Y[29X[2XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(1, H(a,a'))[123X[133X
  
  [33X[0;0YThe  argument is a morphism [23X\alpha: a \rightarrow a'[123X in [23XC[123X. The output is the
  corresponding  morphism  [23X\nu(  \alpha  ):  1 \rightarrow H(a,a')[123X in [23XD[123X of the
  homomorphism structure.[133X
  
  [1X3.16-6 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects[102X( [3Xdistinguished_object[103X, [3Xalpha[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(1, r)[123X[133X
  
  [33X[0;0YThe  arguments  are  the  distinguished  object  [23X1[123X,  a  morphism  [23X\alpha:  a
  \rightarrow  a'[123X, and the object [23Xr = H(a,a')[123X. The output is the corresponding
  morphism [23X\nu( \alpha ): 1 \rightarrow r[123X in [23XD[123X of the homomorphism structure.[133X
  
  [1X3.16-7 InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism[101X
  
  [33X[1;0Y[29X[2XInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism[102X( [3Xa[103X, [3Xa'[103X, [3Xiota[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{C}(a,a')[123X[133X
  
  [33X[0;0YThe  arguments  are  objects  [23Xa,a'[123X  in [23XC[123X and a morphism [23X\iota: 1 \rightarrow
  H(a,a')[123X  in  [23XD[123X.  The output is the corresponding morphism [23X\nu^{-1}(\iota): a
  \rightarrow a'[123X in [23XC[123X of the homomorphism structure.[133X
  
  [1X3.16-8 SolveLinearSystemInAbCategory[101X
  
  [33X[1;0Y[29X[2XSolveLinearSystemInAbCategory[102X( [3Xalpha[103X, [3Xbeta[103X, [3Xgamma[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of morphisms [23X[X_1, \dots, X_n][123X[133X
  
  [33X[0;0YThe  arguments  are  three  lists  [23X\alpha[123X, [23X\beta[123X, and [23X\gamma[123X. The first list
  [23X\alpha[123X  (the  left coefficients) is a list of list of morphisms [23X\alpha_{ij}:
  A_i  \rightarrow B_j[123X, where [23Xi = 1 \dots m[123X and [23Xj = 1 \dots n[123X for integers [23Xm,n
  \geq  1[123X. The second list [23X\beta[123X (the right coefficients) is a list of list of
  morphisms  [23X\beta_{ij}:  C_j  \rightarrow  D_i[123X, where [23Xi = 1 \dots m[123X and [23Xj = 1
  \dots  n[123X.  The  third  list  [23X\gamma[123X  (the right side) is a list of morphisms
  [23X\gamma_i:  A_i  \rightarrow  D_i[123X,  where  [23Xi  =  1,  \dots, m[123X. Assumes that a
  solution to the linear system defined by [23X\alpha[123X, [23X\beta[123X, [23X\gamma[123X exists, i.e.,
  there  exist  morphisms  [23XX_j:  B_j  \rightarrow C_j[123X for [23Xj=1\dots n[123X such that
  [23X\sum_{j  = 1}^n \alpha_{ij}\cdot X_j \cdot \beta_{ij} = \gamma_i[123X for all [23Xi =
  1 \dots m[123X. The output is list of such morphisms [23XX_j: B_j \rightarrow C_j[123X for
  [23Xj=1\dots n[123X.[133X
  
  [1X3.16-9 SolveLinearSystemInAbCategoryOrFail[101X
  
  [33X[1;0Y[29X[2XSolveLinearSystemInAbCategoryOrFail[102X( [3Xalpha[103X, [3Xbeta[103X, [3Xgamma[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of morphisms [23X[X_1, \dots, X_n][123X or [10Xfail[110X[133X
  
  [33X[0;0YLike  [10XSolveLinearSystemInAbCategory[110X,  but  without  the  assumption  that  a
  solution exists. If no solution exists, [10Xfail[110X is returned.[133X
  
  [1X3.16-10 MereExistenceOfSolutionOfLinearSystemInAbCategory[101X
  
  [33X[1;0Y[29X[2XMereExistenceOfSolutionOfLinearSystemInAbCategory[102X( [3Xalpha[103X, [3Xbeta[103X, [3Xgamma[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YLike  [10XSolveLinearSystemInAbCategory[110X,  but  the  output  is  simply [10Xtrue[110X if a
  solution exists, [10Xfalse[110X otherwise.[133X
  
  [1X3.16-11 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(H(a',b), H(a,b'))[123X[133X
  
  [33X[0;0YThis  is  a  convenience  method.  The arguments are two morphisms [23X\alpha: a
  \rightarrow   a',   \beta:   b   \rightarrow   b'[123X   in   [23XC[123X.  The  output  is
  [10XHomomorphismStructureOnMorphisms[110X called on [23X\alpha[123X, [23X\beta[123X.[133X
  
  [1X3.16-12 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xalpha[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(H(a',b), H(a,b))[123X[133X
  
  [33X[0;0YThis  is  a  convenience  method.  The  arguments  are  a morphism [23X\alpha: a
  \rightarrow    a'[123X    and    an    object    [23Xb[123X    in   [23XC[123X.   The   output   is
  [10XHomomorphismStructureOnMorphisms[110X called on [23X\alpha[123X, [23X\mathrm{id}_b[123X.[133X
  
  [1X3.16-13 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xa[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{D}(H(a,b), H(a,b'))[123X[133X
  
  [33X[0;0YThis  is  a convenience method. The arguments are an object [23Xa[123X and a morphism
  [23X\beta: b \rightarrow b'[123X in [23XC[123X. The output is [10XHomomorphismStructureOnMorphisms[110X
  called on [23X\mathrm{id}_a[123X, [23X\beta[123X.[133X
  
  [1X3.16-14 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThis  is  a  convenience method. The arguments are two objects [23Xa[123X and [23Xb[123X in [23XC[123X.
  The output is [10XHomomorphismStructureOnObjects[110X called on [23Xa,b[123X.[133X
  
  [1X3.16-15 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xarg[103X ) [32X operation[133X
  
  [33X[0;0YThis          is          a          convenience          method         for
  [10XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure[110X.[133X
  
  [1X3.16-16 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xarg1[103X, [3Xarg2[103X, [3Xarg3[103X ) [32X operation[133X
  
  [33X[0;0YThis          is          a          convenience          method         for
  [10XInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism[110X.[133X
  
  [1X3.16-17 HomStructure[101X
  
  [33X[1;0Y[29X[2XHomStructure[102X( [3Xarg[103X ) [32X operation[133X
  
  [33X[0;0YThis is a convenience method for [10XDistinguishedObjectOfHomomorphismStructure[110X.[133X
  
  [1X3.16-18 ExtendRangeOfHomomorphismStructureByFullEmbedding[101X
  
  [33X[1;0Y[29X[2XExtendRangeOfHomomorphismStructureByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xobject_function[103X, [3Xmorphism_function[103X, [3Xobject_function_inverse[103X, [3Xmorphism_function_inverse[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHomomorphismStructureOnObjectsExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHomomorphismStructureOnMorphismsExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHomomorphismStructureOnMorphismsWithGivenObjectsExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xs[103X, [3Xalpha[103X, [3Xbeta[103X, [3Xr[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDistinguishedObjectOfHomomorphismStructureExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xalpha[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjectsExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xdistinguished_object[103X, [3Xalpha[103X, [3Xr[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphismExtendedByFullEmbedding[102X( [3XC[103X, [3XE[103X, [3Xa[103X, [3Xa'[103X, [3Xiota[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YIf   [23X\iota\colon   D  \to  E[123X  is  a  full  embedding  of  categories,  every
  [23XD[123X-homomorphism  structure  for  a  category  [23XC[123X  extends  to a [23XE[123X-homomorphism
  structure  for  [23XC[123X.  This  operations  accepts  four  functions  and installs
  operations
  [10XDistinguishedObjectOfHomomorphismStructureExtendedByFullEmbedding[110X,
  [10XHomomorphismStructureOnObjectsExtendedByFullEmbedding[110X  etc. which correspond
  to  the  [23XE[123X-homomorphism  structure for [23XC[123X. Note: To distinguish embeddings in
  different  categories,  in addition to [23XC[123X also [23XE[123X is passed to the operations.
  When  using  this  with different embeddings with the range category [23XE[123X, only
  the last embedding will be used. The arguments are:[133X
  
  [30X    [33X[0;6Y[10Xobject_function[110X gets the categories [23XC[123X and [23XE[123X and an object in [23XD[123X.[133X
  
  [30X    [33X[0;6Y[10Xmorphism_function[110X  gets  the  categories  [23XC[123X  and  [23XE[123X, an object in [23XE[123X, a
        morphism  in [23XD[123X and another object in [23XE[123X. The objects are the results of
        [10Xobject_function[110X applied to the source and range of the morphism.[133X
  
  [30X    [33X[0;6Y[10Xobject_function_inverse[110X  gets the categories [23XC[123X and [23XE[123X and a morphism in
        [23XE[123X.[133X
  
  [30X    [33X[0;6Y[10Xmorphism_function_inverse[110X gets the categories [23XC[123X and [23XE[123X, an object in [23XD[123X,
        a  morphism  in [23XE[123X and another object in [23XD[123X. The objects are the results
        of  [10Xobject_function_inverse[110X  applied  to  the  source and range of the
        morphism.[133X
  
  [33X[0;0Y[10Xobject_function[110X     and     [10Xmorphism_function[110X    define    the    embedding.
  [10Xobject_function_inverse[110X  and [10Xmorphism_function_inverse[110X define the inverse of
  the embedding on its image.[133X
  
  [1X3.16-19 ExtendRangeOfHomomorphismStructureByIdentityAsFullEmbedding[101X
  
  [33X[1;0Y[29X[2XExtendRangeOfHomomorphismStructureByIdentityAsFullEmbedding[102X( [3XC[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YChooses    the    identity    on    [23XD[123X    as    the    full    embedding   in
  [2XExtendRangeOfHomomorphismStructureByFullEmbedding[102X  ([14X3.16-18[114X). This is useful
  to      handle     this     case     as     a     degenerate     case     of
  [2XExtendRangeOfHomomorphismStructureByFullEmbedding[102X ([14X3.16-18[114X).[133X
  
  [1X3.16-20 BasisOfExternalHom[101X
  
  [33X[1;0Y[29X[2XBasisOfExternalHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of morphisms in [23X\mathrm{Hom}_{C}(a,b)[123X[133X
  
  [33X[0;0YThe arguments are objects [23Xa,b[123X in a [23Xk[123X-linear category [23XC[123X. The output is a list
  [23XL[123X  of  morphisms which is a basis of [23X\mathrm{Hom}_{C}(a,b)[123X in the sense that
  any  given  morphism  [23X\alpha:  a  \to  b[123X can uniquely be written as a linear
  combination       of       [23XL[123X       with       the       coefficients      in
  [10XCoefficientsOfMorphismWithGivenBasisOfExternalHom[110X([23X\alpha,L[123X).[133X
  
  [1X3.16-21 CoefficientsOfMorphismWithGivenBasisOfExternalHom[101X
  
  [33X[1;0Y[29X[2XCoefficientsOfMorphismWithGivenBasisOfExternalHom[102X( [3Xalpha[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of elements in [23Xk[123X[133X
  
  [33X[0;0YThe  arguments are a morphism [23X\alpha: a \to b[123X in a [23Xk[123X-linear category [23XC[123X and a
  list  [3XL[103X[10X=BasisOfExternalHom[110X([23Xa,b[123X).  The  output  is  a list of coefficients of
  [23X\alpha[123X with respect to [23XL[123X.[133X
  
  [1X3.16-22 CoefficientsOfMorphism[101X
  
  [33X[1;0Y[29X[2XCoefficientsOfMorphism[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of elements in [23Xk[123X[133X
  
  [33X[0;0YThis  is a convenience method. The argument is a morphism [23X\alpha: a \to b[123X in
  a  [23Xk[123X-linear  category [23XC[123X. The output is a list of coefficients of [23X\alpha[123X with
  respect to the list [10XBasisOfExternalHom[110X([3Xa[103X,[3Xb[103X).[133X
  
  
  [1X3.17 [33X[0;0YSimplified Morphisms[133X[101X
  
  [33X[0;0YLet [23X\phi: A \rightarrow B[123X be a morphism. There are several different natural
  ways to look at [23X\phi[123X as an object in an ambient category:[133X
  
  [30X    [33X[0;6Y[23X\mathrm{Hom}(  A,  B  )[123X, the set of homomorphisms with the equivalence
        relation [23X\mathtt{IsCongruentForMorphisms}[123X regarded as a category,[133X
  
  [30X    [33X[0;6Y[23X\sum_{A}\mathrm{Hom}(  A,  B  )[123X,  the  category of morphisms where the
        range is fixed,[133X
  
  [30X    [33X[0;6Y[23X\sum_{B}\mathrm{Hom}(  A,  B  )[123X,  the  category of morphisms where the
        source is fixed,[133X
  
  [30X    [33X[0;6Y[23X\sum_{A,B}\mathrm{Hom}(  A,  B  )[123X,  the  category  of  morphisms where
        neither source nor range is fixed,[133X
  
  [33X[0;0Yand  furthermore,  if [23X\phi[123X happens to be an endomorphism [23XA \rightarrow A[123X, we
  also have[133X
  
  [30X    [33X[0;6Y[23X\sum_{A}\mathrm{Hom}(A,A)[123X, the category of endomorphisms.[133X
  
  [33X[0;0YLet [23X\mathbf{C}[123X be one of the categories above in which [23X\phi[123X may reside as an
  object, and let [23Xi[123X be a non-negative integer or [23X\infty[123X. CAP provides commands
  for  passing  from  [23X\phi[123X  to  [23X\phi_i[123X,  where [23X\phi_i[123X is isomorphic to [23X\phi[123X in
  [23X\mathbf{C}[123X, but "simpler". The idea is that the greater the [23Xi[123X, the "simpler"
  the  [23X\phi_i[123X (but this could mean the harder the computation), with [23X\infty[123X as
  a  possible  value. The case [23Xi = 0[123X defaults to the identity operator for all
  simplifications.  For  the Add-operatations, only the cases [23Xi \geq 1[123X have to
  be given as functions.[133X
  
  [33X[0;0Y[23X\ [123X[133X
  
  [33X[0;0YIf  we regard [23X\phi[123X as an object in the category [23X\mathrm{Hom}( A, B )[123X, [23X\phi_i[123X
  is again in [23X\mathrm{Hom}( A, B )[123X such that [23X\phi \sim_{A,B} \phi_i[123X. This case
  is handled by the following commands:[133X
  
  [1X3.17-1 SimplifyMorphism[101X
  
  [33X[1;0Y[29X[2XSimplifyMorphism[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,B)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer [23Xi[123X or [10Xinfinity[110X. The output is a simplified morphism [23X\phi_i[123X.[133X
  
  [33X[0;0Y[23X\ [123X[133X
  
  [33X[0;0YIf we regard [23X\phi[123X as an object in the category [23X\sum_{A}\mathrm{Hom}( A, B )[123X,
  then  [23X\phi_i[123X  is  a  morphism  of  type  [23XA_i  \rightarrow  B[123X and there is an
  isomorphism  [23X\sigma_i:  A  \rightarrow  A_i[123X  such that [23X\phi_i \circ \sigma_i
  \sim_{A,B} \phi[123X. This case is handled by the following commands:[133X
  
  [1X3.17-2 SimplifySource[101X
  
  [33X[1;0Y[29X[2XSimplifySource[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,B)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is a simplified morphism with simplified
  source [23X\phi_i: A_i \rightarrow B[123X.[133X
  
  [1X3.17-3 SimplifySource_IsoToInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifySource_IsoToInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,A)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or  [10Xinfinity[110X. The output is the isomorphism [23X(\sigma_i)^{-1}: A_i
  \rightarrow A[123X.[133X
  
  [1X3.17-4 SimplifySource_IsoFromInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifySource_IsoFromInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,A_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer [23Xi[123X or [10Xinfinity[110X. The output is the isomorphism [23X\sigma_i: A \rightarrow
  A_i[123X.[133X
  
  [33X[0;0Y[23X\ [123X[133X
  
  [33X[0;0YIf we regard [23X\phi[123X as an object in the category [23X\sum_{B}\mathrm{Hom}( A, B )[123X,
  then  [23X\phi_i[123X  is  a  morphism  of  type  [23XA  \rightarrow  B_i[123X and there is an
  isomorphism   [23X\rho_i:  B  \rightarrow  B_i[123X  such  that  [23X  \rho_i^{-1}  \circ
  \phi_i\sim_{A,B} \phi[123X. This case is handled by the following commands:[133X
  
  [1X3.17-5 SimplifyRange[101X
  
  [33X[1;0Y[29X[2XSimplifyRange[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,B_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is a simplified morphism with simplified
  range [23X\phi_i: A \rightarrow B_i[123X.[133X
  
  [1X3.17-6 SimplifyRange_IsoToInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifyRange_IsoToInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B_i,B)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or  [10Xinfinity[110X.  The  output is the isomorphism [23X(\rho_i)^{-1}: B_i
  \rightarrow B[123X.[133X
  
  [1X3.17-7 SimplifyRange_IsoFromInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifyRange_IsoFromInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B,B_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is the isomorphism [23X\rho_i: B \rightarrow
  B_i[123X.[133X
  
  [33X[0;0Y[23X\ [123X[133X
  
  [33X[0;0YIf we regard [23X\phi[123X as an object in the category [23X\sum_{A, B}\mathrm{Hom}( A, B
  )[123X,  then  [23X\phi_i[123X  is a morphism of type [23XA_i \rightarrow B_i[123X and there is are
  isomorphisms  [23X\sigma_i: A \rightarrow A_i[123X and [23X\rho_i: B \rightarrow B_i[123X such
  that  [23X \rho_i^{-1} \circ \phi_i \circ \sigma_i \sim_{A,B} \phi[123X. This case is
  handled by the following commands:[133X
  
  [1X3.17-8 SimplifySourceAndRange[101X
  
  [33X[1;0Y[29X[2XSimplifySourceAndRange[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,B_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is a simplified morphism with simplified
  source and range [23X\phi_i: A_i \rightarrow B_i[123X.[133X
  
  [1X3.17-9 SimplifySourceAndRange_IsoToInputRange[101X
  
  [33X[1;0Y[29X[2XSimplifySourceAndRange_IsoToInputRange[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B_i,B)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or  [10Xinfinity[110X.  The  output is the isomorphism [23X(\rho_i)^{-1}: B_i
  \rightarrow B[123X.[133X
  
  [1X3.17-10 SimplifySourceAndRange_IsoFromInputRange[101X
  
  [33X[1;0Y[29X[2XSimplifySourceAndRange_IsoFromInputRange[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B,B_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is the isomorphism [23X\rho_i: B \rightarrow
  B_i[123X.[133X
  
  [1X3.17-11 SimplifySourceAndRange_IsoToInputSource[101X
  
  [33X[1;0Y[29X[2XSimplifySourceAndRange_IsoToInputSource[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,A)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer  [23Xi[123X  or  [10Xinfinity[110X. The output is the isomorphism [23X(\sigma_i)^{-1}: A_i
  \rightarrow A[123X.[133X
  
  [1X3.17-12 SimplifySourceAndRange_IsoFromInputSource[101X
  
  [33X[1;0Y[29X[2XSimplifySourceAndRange_IsoFromInputSource[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,A_i)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  morphism  [23X\phi:  A  \rightarrow B[123X and a non-negative
  integer [23Xi[123X or [10Xinfinity[110X. The output is the isomorphism [23X\sigma_i: A \rightarrow
  A_i[123X.[133X
  
  [33X[0;0Y[23X\ [123X[133X
  
  [33X[0;0YIf [23X\phi:A \rightarrow A[123X is an endomorphism, we may regard it as an object in
  the category [23X\sum_{A}\mathrm{Hom}( A, A )[123X. In this case [23X\phi_i[123X is a morphism
  of  type  [23XA_i  \rightarrow  A_i[123X  and  there  is  an  isomorphism [23X\sigma_i: A
  \rightarrow  A_i[123X  such  that  [23X  \sigma_i^{-1}  \circ  \phi_i  \circ \sigma_i
  \sim_{A,A} \phi[123X. This case is handled by the following commands:[133X
  
  [1X3.17-13 SimplifyEndo[101X
  
  [33X[1;0Y[29X[2XSimplifyEndo[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,A_i)[123X[133X
  
  [33X[0;0YThe  arguments  are an endomorphism [23X\phi: A \rightarrow A[123X and a non-negative
  integer  [23Xi[123X  or [10Xinfinity[110X. The output is a simplified endomorphism [23X\phi_i: A_i
  \rightarrow A_i[123X.[133X
  
  [1X3.17-14 SimplifyEndo_IsoToInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifyEndo_IsoToInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_i,A)[123X[133X
  
  [33X[0;0YThe  arguments  are an endomorphism [23X\phi: A \rightarrow A[123X and a non-negative
  integer  [23Xi[123X  or  [10Xinfinity[110X. The output is the isomorphism [23X(\sigma_i)^{-1}: A_i
  \rightarrow A[123X.[133X
  
  [1X3.17-15 SimplifyEndo_IsoFromInputObject[101X
  
  [33X[1;0Y[29X[2XSimplifyEndo_IsoFromInputObject[102X( [3Xphi[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,A_i)[123X[133X
  
  [33X[0;0YThe  arguments  are an endomorphism [23X\phi: A \rightarrow A[123X and a non-negative
  integer [23Xi[123X or [10Xinfinity[110X. The output is the isomorphism [23X\sigma_i: A \rightarrow
  A_i[123X.[133X
  
  [1X3.17-16 Simplify[101X
  
  [33X[1;0Y[29X[2XSimplify[102X( [3Xphi[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A_{\infty},B_{\infty})[123X[133X
  
  [33X[0;0YThis  is a convenient method. The argument is a morphism [23X\phi: A \rightarrow
  B[123X.  The  output is a "simplified" version of [23X\phi[123X that may change the source
  and  range  of  [23X\phi[123X  (up  to  isomorphism). To be precise, the output is an
  [23X\infty[123X-th  simplified  morphism  of  [23X(\iota_A^{\infty})^{-1}\circ \phi \circ
  \iota_A^{\infty}[123X.[133X
  
  
  [1X3.18 [33X[0;0YReduction by split epi summands[133X[101X
  
  [33X[0;0YLet  [23X\alpha:  A \rightarrow B[123X be a morphism in an additive category. Suppose
  we  are  given  direct  sum  decompositions  of [23XA \simeq A' \oplus A''[123X and [23XB
  \simeq  B'  \oplus B''[123X such that If [23X\alpha''[123X is a split epimorphism, then we
  call  [23X\alpha':  A'  \rightarrow  B'[123X  [13Xsome  reduction  of [23X\alpha[123X by split epi
  summands[113X.  The inclusions/projections of the decompositions into direct sums
  induce commutative diagrams and[133X
  
  [1X3.18-1 SomeReductionBySplitEpiSummand[101X
  
  [33X[1;0Y[29X[2XSomeReductionBySplitEpiSummand[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A',B')[123X[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  A \rightarrow B[123X. The output is some
  reduction of [23X\alpha[123X by split epi summands [23X\alpha': A' \rightarrow B'[123X.[133X
  
  [1X3.18-2 SomeReductionBySplitEpiSummand_MorphismToInputRange[101X
  
  [33X[1;0Y[29X[2XSomeReductionBySplitEpiSummand_MorphismToInputRange[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B',B)[123X[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  A  \rightarrow B[123X. The output is the
  morphism  [23X\beta':  B'  \rightarrow  B[123X  linking [23X\alpha[123X with some reduction by
  split epi summands.[133X
  
  [1X3.18-3 SomeReductionBySplitEpiSummand_MorphismFromInputRange[101X
  
  [33X[1;0Y[29X[2XSomeReductionBySplitEpiSummand_MorphismFromInputRange[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(B,B')[123X[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  A  \rightarrow B[123X. The output is the
  morphism [23X\beta: B \rightarrow B'[123X linking [23X\alpha[123X with some reduction by split
  epi summands.[133X
  
