  
  [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
  
  [29X[2XCapCategory[102X( [3Xalpha[103X ) [32X attribute
  [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
  
  [29X[2XSource[102X( [3Xalpha[103X ) [32X attribute
  [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
  
  [29X[2XRange[102X( [3Xalpha[103X ) [32X attribute
  [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;0YCategorical Properties of Morphisms[133X[101X
  
  [1X3.2-1 AddIsMonomorphism[101X
  
  [29X[2XAddIsMonomorphism[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 [10XIsMonomorphism[110X. [23XF:
  \alpha \mapsto \mathtt{IsMonomorphism}(\alpha)[123X.[133X
  
  [1X3.2-2 AddIsEpimorphism[101X
  
  [29X[2XAddIsEpimorphism[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 [10XIsEpimorphism[110X. [23XF:
  \alpha \mapsto \mathtt{IsEpimorphism}(\alpha)[123X.[133X
  
  [1X3.2-3 AddIsIsomorphism[101X
  
  [29X[2XAddIsIsomorphism[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 [10XIsIsomorphism[110X. [23XF:
  \alpha \mapsto \mathtt{IsIsomorphism}(\alpha)[123X.[133X
  
  [1X3.2-4 AddIsSplitMonomorphism[101X
  
  [29X[2XAddIsSplitMonomorphism[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
  [10XIsSplitMonomorphism[110X. [23XF: \alpha \mapsto \mathtt{IsSplitMonomorphism}(\alpha)[123X.[133X
  
  [1X3.2-5 AddIsSplitEpimorphism[101X
  
  [29X[2XAddIsSplitEpimorphism[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 [10XIsSplitEpimorphism[110X.
  [23XF: \alpha \mapsto \mathtt{IsSplitEpimorphism}(\alpha)[123X.[133X
  
  [1X3.2-6 AddIsOne[101X
  
  [29X[2XAddIsOne[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 [10XIsOne[110X. [23XF: \alpha
  \mapsto \mathtt{IsOne}(\alpha)[123X.[133X
  
  [1X3.2-7 AddIsIdempotent[101X
  
  [29X[2XAddIsIdempotent[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 [10XIsIdempotent[110X. [23XF:
  \alpha \mapsto \mathtt{IsIdempotent}(\alpha)[123X.[133X
  
  
  [1X3.3 [33X[0;0YNon-Categorical Properties of Morphisms[133X[101X
  
  [33X[0;0YNon-categorical properties are not stable under equivalences of categories.[133X
  
  [1X3.3-1 IsIdenticalToIdentityMorphism[101X
  
  [29X[2XIsIdenticalToIdentityMorphism[102X( [3Xalpha[103X ) [32X property
  [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
  
  [1X3.3-2 AddIsIdenticalToIdentityMorphism[101X
  
  [29X[2XAddIsIdenticalToIdentityMorphism[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
  [10XIsIdenticalToIdentityMorphism[110X.           [23XF:          \alpha          \mapsto
  \mathtt{IsIdenticalToIdentityMorphism}(\alpha)[123X.[133X
  
  [1X3.3-3 IsIdenticalToZeroMorphism[101X
  
  [29X[2XIsIdenticalToZeroMorphism[102X( [3Xalpha[103X ) [32X property
  [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
  
  [1X3.3-4 AddIsIdenticalToZeroMorphism[101X
  
  [29X[2XAddIsIdenticalToZeroMorphism[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
  [10XIsIdenticalToZeroMorphism[110X.            [23XF:            \alpha           \mapsto
  \mathtt{IsIdenticalToZeroMorphism }(\alpha)[123X.[133X
  
  [1X3.3-5 AddIsEndomorphism[101X
  
  [29X[2XAddIsEndomorphism[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 [10XIsEndomorphism[110X. [23XF:
  \alpha \mapsto \mathtt{IsEndomorphism}(\alpha)[123X.[133X
  
  [1X3.3-6 AddIsAutomorphism[101X
  
  [29X[2XAddIsAutomorphism[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 [10XIsAutomorphism[110X. [23XF:
  \alpha \mapsto \mathtt{IsAutomorphism}(\alpha)[123X.[133X
  
  
  [1X3.4 [33X[0;0YEquality and Congruence for Morphisms[133X[101X
  
  [1X3.4-1 IsCongruentForMorphisms[101X
  
  [29X[2XIsCongruentForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.4-2 AddIsCongruentForMorphisms[101X
  
  [29X[2XAddIsCongruentForMorphisms[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
  [10XIsCongruentForMorphisms[110X.        [23XF:       (\alpha,       \beta)       \mapsto
  \mathtt{IsCongruentForMorphisms}(\alpha, \beta)[123X.[133X
  
  [1X3.4-3 IsEqualForMorphisms[101X
  
  [29X[2XIsEqualForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.4-4 AddIsEqualForMorphisms[101X
  
  [29X[2XAddIsEqualForMorphisms[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
  [10XIsEqualForMorphisms[110X.         [23XF:        (\alpha,        \beta)        \mapsto
  \mathtt{IsEqualForMorphisms}(\alpha, \beta)[123X.[133X
  
  [1X3.4-5 IsEqualForMorphismsOnMor[101X
  
  [29X[2XIsEqualForMorphismsOnMor[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.4-6 AddIsEqualForMorphismsOnMor[101X
  
  [29X[2XAddIsEqualForMorphismsOnMor[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
  [10XIsEqualForMorphismsOnMor[110X.       [23XF:       (\alpha,       \beta)       \mapsto
  \mathtt{IsEqualForMorphismsOnMor}(\alpha, \beta)[123X.[133X
  
  
  [1X3.5 [33X[0;0YBasic Operations for Morphisms in Ab-Categories[133X[101X
  
  [1X3.5-1 IsZeroForMorphisms[101X
  
  [29X[2XIsZeroForMorphisms[102X( [3Xalpha[103X ) [32X operation
  [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
  
  [1X3.5-2 AddIsZeroForMorphisms[101X
  
  [29X[2XAddIsZeroForMorphisms[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 [10XIsZeroForMorphisms[110X.
  [23XF: \alpha \mapsto \mathtt{IsZeroForMorphisms}(\alpha)[123X.[133X
  
  [1X3.5-3 AdditionForMorphisms[101X
  
  [29X[2XAdditionForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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.[133X
  
  [1X3.5-4 AddAdditionForMorphisms[101X
  
  [29X[2XAddAdditionForMorphisms[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
  [10XAdditionForMorphisms[110X. [23XF: (\alpha, \beta) \mapsto \alpha + \beta[123X.[133X
  
  [1X3.5-5 SubtractionForMorphisms[101X
  
  [29X[2XSubtractionForMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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.[133X
  
  [1X3.5-6 AddSubtractionForMorphisms[101X
  
  [29X[2XAddSubtractionForMorphisms[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
  [10XSubtractionForMorphisms[110X. [23XF: (\alpha, \beta) \mapsto \alpha - \beta[123X.[133X
  
  [1X3.5-7 AdditiveInverseForMorphisms[101X
  
  [29X[2XAdditiveInverseForMorphisms[102X( [3Xalpha[103X ) [32X operation
  [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.[133X
  
  [1X3.5-8 AddAdditiveInverseForMorphisms[101X
  
  [29X[2XAddAdditiveInverseForMorphisms[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
  [10XAdditiveInverseForMorphisms[110X. [23XF: \alpha \mapsto -\alpha[123X.[133X
  
  [1X3.5-9 ZeroMorphism[101X
  
  [29X[2XZeroMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation
  [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
  
  [1X3.5-10 AddZeroMorphism[101X
  
  [29X[2XAddZeroMorphism[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 [10XZeroMorphism[110X. [23XF:
  (a,b) \mapsto (0: a \rightarrow b)[123X.[133X
  
  
  [1X3.6 [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
  
  [1X3.6-1 IsEqualAsSubobjects[101X
  
  [29X[2XIsEqualAsSubobjects[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.6-2 AddIsEqualAsSubobjects[101X
  
  [29X[2XAddIsEqualAsSubobjects[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
  [10XIsEqualAsSubobjects[110X.         [23XF:        (\alpha,        \beta)        \mapsto
  \mathtt{IsEqualAsSubobjects}(\alpha,\beta)[123X.[133X
  
  [1X3.6-3 IsEqualAsFactorobjects[101X
  
  [29X[2XIsEqualAsFactorobjects[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.6-4 AddIsEqualAsFactorobjects[101X
  
  [29X[2XAddIsEqualAsFactorobjects[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
  [10XIsEqualAsFactorobjects[110X.        [23XF:        (\alpha,       \beta)       \mapsto
  \mathtt{IsEqualAsFactorobjects}(\alpha,\beta)[123X.[133X
  
  [1X3.6-5 IsDominating[101X
  
  [29X[2XIsDominating[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YIn  short:  Returns  [10Xtrue[110X  iff  [23X\alpha[123X  is  smaller  than  [23X\beta[123X.  [23X\\  [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
  
  [1X3.6-6 AddIsDominating[101X
  
  [29X[2XAddIsDominating[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 [10XIsDominating[110X. [23XF:
  (\alpha, \beta) \mapsto \mathtt{IsDominating}(\alpha,\beta)[123X.[133X
  
  [1X3.6-7 IsCodominating[101X
  
  [29X[2XIsCodominating[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YIn  short:  Returns  [10Xtrue[110X  iff  [23X\alpha[123X  is  smaller  than  [23X\beta[123X.  [23X\\  [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
  
  [1X3.6-8 AddIsCodominating[101X
  
  [29X[2XAddIsCodominating[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 [10XIsCodominating[110X. [23XF:
  (\alpha, \beta) \mapsto \mathtt{IsCodominating}(\alpha,\beta)[123X.[133X
  
  
  [1X3.7 [33X[0;0YIdentity Morphism and Composition of Morphisms[133X[101X
  
  [1X3.7-1 IdentityMorphism[101X
  
  [29X[2XIdentityMorphism[102X( [3Xa[103X ) [32X attribute
  [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
  
  [1X3.7-2 AddIdentityMorphism[101X
  
  [29X[2XAddIdentityMorphism[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 [10XIdentityMorphism[110X.
  [23XF: a \mapsto \mathrm{id}_a[123X.[133X
  
  [1X3.7-3 PreCompose[101X
  
  [29X[2XPreCompose[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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
  
  [1X3.7-4 PreCompose[101X
  
  [29X[2XPreCompose[102X( [3XL[103X ) [32X operation
  [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
  
  [1X3.7-5 AddPreCompose[101X
  
  [29X[2XAddPreCompose[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 [10XPreCompose[110X. [23XF:
  (\alpha, \beta) \mapsto \beta \circ \alpha[123X.[133X
  
  [1X3.7-6 PostCompose[101X
  
  [29X[2XPostCompose[102X( [3Xbeta[103X, [3Xalpha[103X ) [32X operation
  [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
  
  [1X3.7-7 PostCompose[101X
  
  [29X[2XPostCompose[102X( [3XL[103X ) [32X operation
  [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
  
  [1X3.7-8 AddPostCompose[101X
  
  [29X[2XAddPostCompose[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 [10XPostCompose[110X. [23XF:
  (\alpha, \beta) \mapsto \alpha \circ \beta[123X.[133X
  
  
  [1X3.8 [33X[0;0YWell-Definedness of Morphisms[133X[101X
  
  [1X3.8-1 IsWellDefinedForMorphisms[101X
  
  [29X[2XIsWellDefinedForMorphisms[102X( [3Xalpha[103X ) [32X operation
  [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
  
  [1X3.8-2 AddIsWellDefinedForMorphisms[101X
  
  [29X[2XAddIsWellDefinedForMorphisms[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
  [10XIsWellDefinedForMorphisms[110X.            [23XF:            \alpha           \mapsto
  \mathtt{IsWellDefinedForMorphisms}( \alpha )[123X.[133X
  
  
  [1X3.9 [33X[0;0YBasic Operations for Morphisms in Abelian Categories[133X[101X
  
  [1X3.9-1 LiftAlongMonomorphism[101X
  
  [29X[2XLiftAlongMonomorphism[102X( [3Xiota[103X, [3Xtau[103X ) [32X operation
  [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
  
  [1X3.9-2 AddLiftAlongMonomorphism[101X
  
  [29X[2XAddLiftAlongMonomorphism[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
  [10XLiftAlongMonomorphism[110X.  The function [23XF[123X maps a pair [23X(\iota, \tau)[123X to a lift [23Xu[123X
  if it exists, and to [10Xfail[110X otherwise.[133X
  
  [1X3.9-3 ColiftAlongEpimorphism[101X
  
  [29X[2XColiftAlongEpimorphism[102X( [3Xepsilon[103X, [3Xtau[103X ) [32X operation
  [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
  
  [1X3.9-4 AddColiftAlongEpimorphism[101X
  
  [29X[2XAddColiftAlongEpimorphism[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
  [10XColiftAlongEpimorphism[110X.  The  function  [23XF[123X  maps a pair [23X(\epsilon, \tau)[123X to a
  lift [23Xu[123X if it exists, and to [10Xfail[110X otherwise.[133X
  
  
  [1X3.10 [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
  
  [1X3.10-1 Lift[101X
  
  [29X[2XLift[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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  such that there is 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. The output is such a lift or [23X\mathtt{fail}[123X if it doesn't
  exist.[133X
  
  [1X3.10-2 AddLift[101X
  
  [29X[2XAddLift[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 [10XLift[110X. The function
  [23XF[123X  maps a pair [23X(\alpha, \beta)[123X to a lift [23X\alpha / \beta[123X if it exists, and to
  [10Xfail[110X otherwise.[133X
  
  [1X3.10-3 Colift[101X
  
  [29X[2XColift[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation
  [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  such  that  there  is  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. The output is such a colift
  or [23X\mathtt{fail}[123X if it doesn't exist.[133X
  
  [1X3.10-4 AddColift[101X
  
  [29X[2XAddColift[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 [10XColift[110X. The
  function  [23XF[123X  maps a pair [23X(\alpha, \beta)[123X to a colift [23X\alpha \backslash \beta[123X
  if it exists, and to [10Xfail[110X otherwise.[133X
  
  
  [1X3.11 [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
  
  [1X3.11-1 AddInverse[101X
  
  [29X[2XAddInverse[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 [10XInverse[110X. [23XF: \alpha
  \mapsto \alpha^{-1}[123X.[133X
  
  
  [1X3.12 [33X[0;0YTool functions for caches[133X[101X
  
  [1X3.12-1 IsEqualForCacheForMorphisms[101X
  
  [29X[2XIsEqualForCacheForMorphisms[102X( [3Xphi[103X, [3Xpsi[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YCompares two objects in the cache[133X
  
  [1X3.12-2 AddIsEqualForCacheForMorphisms[101X
  
  [29X[2XAddIsEqualForCacheForMorphisms[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 morphisms in the category,
  IsEqualForCacheForMorphism  is  used.  By  default  this  is  an  alias  for
  IsEqualForMorphismsOnMor,  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
  
