  
  [1X4 [33X[0;0YGeneralized Morphism Category by Three Arrows[133X[101X
  
  
  [1X4.1 [33X[0;0YGAP Categories[133X[101X
  
  [1X4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject[101X
  
  [33X[1;0Y[29X[2XIsGeneralizedMorphismCategoryByThreeArrowsObject[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe  GAP  category  of objects in the generalized morphism category by three
  arrows.[133X
  
  [1X4.1-2 IsGeneralizedMorphismByThreeArrows[101X
  
  [33X[1;0Y[29X[2XIsGeneralizedMorphismByThreeArrows[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe  GAP category of morphisms in the generalized morphism category by three
  arrows.[133X
  
  
  [1X4.2 [33X[0;0YProperties[133X[101X
  
  [1X4.2-1 HasIdentitiesAsReversedArrows[101X
  
  [33X[1;0Y[29X[2XHasIdentitiesAsReversedArrows[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow  c  \leftarrow  d[123X.  The  output  is [10Xtrue[110X if [23Xa \leftarrow b[123X and [23Xc
  \leftarrow d[123X are congruent to identity morphisms, [10Xfalse[110X otherwise.[133X
  
  [1X4.2-2 HasIdentityAsSourceAid[101X
  
  [33X[1;0Y[29X[2XHasIdentityAsSourceAid[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow  c  \leftarrow  d[123X.  The  output  is  [10Xtrue[110X  if  [23Xa \leftarrow b[123X is
  congruent to an identity morphism, [10Xfalse[110X otherwise.[133X
  
  [1X4.2-3 HasIdentityAsRangeAid[101X
  
  [33X[1;0Y[29X[2XHasIdentityAsRangeAid[102X( [3Xalpha[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow  c  \leftarrow  d[123X.  The  output  is  [10Xtrue[110X  if  [23Xc \leftarrow d[123X is
  congruent to an identity morphism, [10Xfalse[110X otherwise.[133X
  
  
  [1X4.3 [33X[0;0YAttributes[133X[101X
  
  [1X4.3-1 UnderlyingHonestObject[101X
  
  [33X[1;0Y[29X[2XUnderlyingHonestObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object in [23X\mathbf{A}[123X[133X
  
  [33X[0;0YThe  argument  is  an object [23Xa[123X in the generalized morphism category by three
  arrows. The output is its underlying honest object.[133X
  
  [1X4.3-2 SourceAid[101X
  
  [33X[1;0Y[29X[2XSourceAid[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{A}}(b,a)[123X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow c \leftarrow d[123X. The output is its source aid [23Xa \leftarrow b[123X.[133X
  
  [1X4.3-3 RangeAid[101X
  
  [33X[1;0Y[29X[2XRangeAid[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{A}}(d,c)[123X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow c \leftarrow d[123X. The output is its range aid [23Xc \leftarrow d[123X.[133X
  
  [1X4.3-4 Arrow[101X
  
  [33X[1;0Y[29X[2XArrow[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{A}}(b,c)[123X[133X
  
  [33X[0;0YThe argument is a generalized morphism [23X\alpha[123X by three arrows [23Xa \leftarrow b
  \rightarrow c \leftarrow d[123X. The output is its range aid [23Xb \rightarrow c[123X.[133X
  
  [1X4.3-5 PseudoInverse[101X
  
  [33X[1;0Y[29X[2XPseudoInverse[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(b,a)[123X[133X
  
  [33X[0;0YThe  argument  is  a  generalized  morphism [23X\alpha: a \rightarrow b[123X by three
  arrows. The output is its pseudo inverse [23Xb \rightarrow a[123X.[133X
  
  [1X4.3-6 GeneralizedInverseByThreeArrows[101X
  
  [33X[1;0Y[29X[2XGeneralizedInverseByThreeArrows[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(b,a)[123X[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  a \rightarrow b \in \mathbf{A}[123X. The
  output is its generalized inverse [23Xb \rightarrow a[123X by three arrows.[133X
  
  [1X4.3-7 IdempotentDefinedBySubobjectByThreeArrows[101X
  
  [33X[1;0Y[29X[2XIdempotentDefinedBySubobjectByThreeArrows[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(b,b)[123X[133X
  
  [33X[0;0YThe  argument is a subobject [23X\alpha: a \hookrightarrow b \in \mathbf{A}[123X. The
  output  is  the idempotent [23Xb \rightarrow b \in \mathbf{G(A)}[123X by three arrows
  defined by [23X\alpha[123X.[133X
  
  [1X4.3-8 IdempotentDefinedByFactorobjectByThreeArrows[101X
  
  [33X[1;0Y[29X[2XIdempotentDefinedByFactorobjectByThreeArrows[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(b,b)[123X[133X
  
  [33X[0;0YThe   argument   is  a  factorobject  [23X\alpha:  b  \twoheadrightarrow  a  \in
  \mathbf{A}[123X.  The  output is the idempotent [23Xb \rightarrow b \in \mathbf{G(A)}[123X
  by three arrows defined by [23X\alpha[123X.[133X
  
  
  [1X4.4 [33X[0;0YOperations[133X[101X
  
  [1X4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismFromFactorToSubobjectByThreeArrows[102X( [3Xbeta[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(c,a)[123X[133X
  
  [33X[0;0YThe  arguments  are  a  a  factorobject [23X\beta: b \twoheadrightarrow c[123X, and a
  subobject  [23X\alpha:  a  \hookrightarrow  b[123X.  The  output  is  the generalized
  morphism by three arrows from the factorobject to the subobject.[133X
  
  [1X4.4-2 CommonCoastriction[101X
  
  [33X[1;0Y[29X[2XCommonCoastriction[102X( [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of generalized morphisms[133X
  
  [33X[0;0YThe argument is a list [23XL[123X of generalized morphisms by three arrows having the
  same  range.  The  output is a list of generalized morphisms by three arrows
  which is the comman coastriction of [23XL[123X.[133X
  
  
  [1X4.5 [33X[0;0YConstructors[133X[101X
  
  [1X4.5-1 GeneralizedMorphismByThreeArrows[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismByThreeArrows[102X( [3Xalpha[103X, [3Xbeta[103X, [3Xgamma[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(a,d)[123X[133X
  
  [33X[0;0YThe  arguments are morphisms [23X\alpha: a \leftarrow b[123X, [23X\beta: b \rightarrow c[123X,
  and  [23X\gamma:  c  \leftarrow  d[123X  in  [23X\mathbf{A}[123X.  The output is a generalized
  morphism  by three arrows with source aid [23X\alpha[123X, arrow [23X\beta[123X, and range aid
  [23X\gamma[123X.[133X
  
  [1X4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismByThreeArrowsWithSourceAid[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(a,c)[123X[133X
  
  [33X[0;0YThe arguments are morphisms [23X\alpha: a \leftarrow b[123X, and [23X\beta: b \rightarrow
  c[123X  in  [23X\mathbf{A}[123X.  The  output  is  a  generalized morphism by three arrows
  defined  by  the composition of the given two arrows regarded as generalized
  morphisms.[133X
  
  [1X4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismByThreeArrowsWithRangeAid[102X( [3Xbeta[103X, [3Xgamma[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(b,d)[123X[133X
  
  [33X[0;0YThe arguments are morphisms [23X\beta: b \rightarrow c[123X, and [23X\gamma: c \leftarrow
  d[123X  in  [23X\mathbf{A}[123X.  The  output  is  a  generalized morphism by three arrows
  defined  by  the composition of the given two arrows regarded as generalized
  morphisms.[133X
  
  [1X4.5-4 AsGeneralizedMorphismByThreeArrows[101X
  
  [33X[1;0Y[29X[2XAsGeneralizedMorphismByThreeArrows[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(a,b)[123X[133X
  
  [33X[0;0YThe argument is a morphism [23X\alpha: a \rightarrow b[123X in [23X\mathbf{A}[123X. The output
  is the honest generalized morphism by three arrows defined by [23X\alpha[123X.[133X
  
  [1X4.5-5 GeneralizedMorphismCategoryByThreeArrows[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismCategoryByThreeArrows[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe   argument  is  an  abelian  category  [23X\mathbf{A}[123X.  The  output  is  its
  generalized morphism category [23X\mathbf{G(A)}[123X by three arrows.[133X
  
  [1X4.5-6 GeneralizedMorphismByThreeArrowsObject[101X
  
  [33X[1;0Y[29X[2XGeneralizedMorphismByThreeArrowsObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object in [23X\mathbf{G(A)}[123X[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X in an abelian category [23X\mathbf{A}[123X. The output is
  the  object  in  the  generalized  morphism  category  by three arrows whose
  underlying honest object is [23Xa[123X.[133X
  
