  
  [1X3 [33X[0;0YGeneralized Morphism Category by Spans[133X[101X
  
  
  [1X3.1 [33X[0;0YGAP Categories[133X[101X
  
  [1X3.1-1 IsGeneralizedMorphismCategoryBySpansObject[101X
  
  [29X[2XIsGeneralizedMorphismCategoryBySpansObject[102X( [3Xobject[103X ) [32X filter
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe GAP category of objects in the generalized morphism category by spans.[133X
  
  [1X3.1-2 IsGeneralizedMorphismBySpan[101X
  
  [29X[2XIsGeneralizedMorphismBySpan[102X( [3Xobject[103X ) [32X filter
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe GAP category of morphisms in the generalized morphism category by spans.[133X
  
  
  [1X3.2 [33X[0;0YProperties[133X[101X
  
  [1X3.2-1 HasIdentityAsReversedArrow[101X
  
  [29X[2XHasIdentityAsReversedArrow[102X( [3Xalpha[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe  argument  is  a  generalized  morphism  [23X\alpha[123X by a span [23Xa \leftarrow b
  \rightarrow  c[123X.  The  output  is  [10Xtrue[110X  if [23Xa \leftarrow b[123X is congruent to an
  identity morphism, [10Xfalse[110X otherwise.[133X
  
  
  [1X3.3 [33X[0;0YAttributes[133X[101X
  
  [1X3.3-1 UnderlyingHonestObject[101X
  
  [29X[2XUnderlyingHonestObject[102X( [3Xa[103X ) [32X attribute
  [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 spans.
  The output is its underlying honest object.[133X
  
  [1X3.3-2 Arrow[101X
  
  [29X[2XArrow[102X( [3Xalpha[103X ) [32X attribute
  [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 a span [23Xa \leftarrow b
  \rightarrow c[123X. The output is its arrow [23Xb \rightarrow c[123X.[133X
  
  [1X3.3-3 ReversedArrow[101X
  
  [29X[2XReversedArrow[102X( [3Xalpha[103X ) [32X attribute
  [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 a span [23Xa \leftarrow b
  \rightarrow c[123X. The output is its reversed arrow [23Xa \leftarrow b[123X.[133X
  
  [1X3.3-4 NormalizedSpanTuple[101X
  
  [29X[2XNormalizedSpanTuple[102X( [3Xalpha[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya pair of morphisms in [23X\mathbf{A}[123X.[133X
  
  [33X[0;0YThe  argument  is  a generalized morphism [23X\alpha: a \rightarrow b[123X by a span.
  The output is its normalized span pair [23X(a \leftarrow d, d \rightarrow b)[123X.[133X
  
  [1X3.3-5 PseudoInverse[101X
  
  [29X[2XPseudoInverse[102X( [3Xalpha[103X ) [32X attribute
  [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 a span.
  The output is its pseudo inverse [23Xb \rightarrow a[123X.[133X
  
  [1X3.3-6 GeneralizedInverseBySpan[101X
  
  [29X[2XGeneralizedInverseBySpan[102X( [3Xalpha[103X ) [32X attribute
  [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 span.[133X
  
  [1X3.3-7 IdempotentDefinedBySubobjectBySpan[101X
  
  [29X[2XIdempotentDefinedBySubobjectBySpan[102X( [3Xalpha[103X ) [32X attribute
  [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 span defined
  by [23X\alpha[123X.[133X
  
  [1X3.3-8 IdempotentDefinedByFactorobjectBySpan[101X
  
  [29X[2XIdempotentDefinedByFactorobjectBySpan[102X( [3Xalpha[103X ) [32X attribute
  [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 span defined by [23X\alpha[123X.[133X
  
  [1X3.3-9 NormalizedSpan[101X
  
  [29X[2XNormalizedSpan[102X( [3Xalpha[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(a,b)[123X[133X
  
  [33X[0;0YThe  argument  is  a generalized morphism [23X\alpha: a \rightarrow b[123X by a span.
  The output is its normalization by span.[133X
  
  
  [1X3.4 [33X[0;0YOperations[133X[101X
  
  [1X3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan[101X
  
  [29X[2XGeneralizedMorphismFromFactorToSubobjectBySpan[102X( [3Xbeta[103X, [3Xalpha[103X ) [32X operation
  [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 span from the factorobject to the subobject.[133X
  
  
  [1X3.5 [33X[0;0YConstructors[133X[101X
  
  [1X3.5-1 GeneralizedMorphismBySpan[101X
  
  [29X[2XGeneralizedMorphismBySpan[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}_{\mathbf{G(A)}}(a,b)[123X[133X
  
  [33X[0;0YThe  arguments are morphisms [23X\alpha: a \leftarrow c[123X and [23X\beta: c \rightarrow
  b[123X  in  [23X\mathbf{A}[123X.  The  output is a generalized morphism by span with arrow
  [23X\beta[123X and reversed arrow [23X\alpha[123X.[133X
  
  [1X3.5-2 GeneralizedMorphismBySpan[101X
  
  [29X[2XGeneralizedMorphismBySpan[102X( [3Xalpha[103X, [3Xbeta[103X, [3Xgamma[103X ) [32X operation
  [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 span defined by the composition the given three arrows regarded
  as generalized morphisms.[133X
  
  [1X3.5-3 GeneralizedMorphismBySpanWithRangeAid[101X
  
  [29X[2XGeneralizedMorphismBySpanWithRangeAid[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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 \rightarrow b[123X, and [23X\beta: b \leftarrow
  c[123X in [23X\mathbf{A}[123X. The output is a generalized morphism by span defined by the
  composition the given two arrows regarded as generalized morphisms.[133X
  
  [1X3.5-4 AsGeneralizedMorphismBySpan[101X
  
  [29X[2XAsGeneralizedMorphismBySpan[102X( [3Xalpha[103X ) [32X attribute
  [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 span defined by [23X\alpha[123X.[133X
  
  [1X3.5-5 GeneralizedMorphismCategoryBySpans[101X
  
  [29X[2XGeneralizedMorphismCategoryBySpans[102X( [3XA[103X ) [32X attribute
  [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 spans.[133X
  
  [1X3.5-6 GeneralizedMorphismBySpansObject[101X
  
  [29X[2XGeneralizedMorphismBySpansObject[102X( [3Xa[103X ) [32X attribute
  [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 spans whose underlying
  honest object is [23Xa[123X.[133X
  
