chap12.txt

  
  12 Examples and Tests
  
  
  12.1 Spectral Sequences
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInSingular( );
    Z
    gap> C1 := FreeLeftPresentation( 1, ZZ );
    <An object in Category of left presentations of Z>
    gap> C2 := FreeLeftPresentation( 2, ZZ );
    <An object in Category of left presentations of Z>
    gap> h1 := PresentationMorphism( C2, HomalgMatrix( [ [ 0 ], [ 4 ] ], ZZ ), C1 );
    <A morphism in Category of left presentations of Z>
    gap> h2 := PresentationMorphism( C2, HomalgMatrix( [ [ 0 ], [ 2 ] ], ZZ ), C1 );
    <A morphism in Category of left presentations of Z>
    gap> v1 := PresentationMorphism( C2, HomalgMatrix( [ [ 2, 0 ], [ 1, 2 ] ], ZZ ), C2 );
    <A morphism in Category of left presentations of Z>
    gap> v2 := PresentationMorphism( C1, HomalgMatrix( [ [ 4 ] ], ZZ ), C1 );
    <A morphism in Category of left presentations of Z>
    gap> cocomplex_h1 := CocomplexFromMorphismList( [ h1 ] );
    <An object in Cocomplex category of Category of left presentations of Z>
    gap> cocomplex_h2 := CocomplexFromMorphismList( [ h2 ] );
    <An object in Cocomplex category of Category of left presentations of Z>
    gap> cocomplex_mor := CochainMap( cocomplex_h2, [ v1, v2 ], cocomplex_h1 );
    <A morphism in Cocomplex category of Category of left presentations of Z>
    gap> Zmod := CapCategory( C1 );
    Category of left presentations of Z
    gap> CH0 := CohomologyFunctor( Zmod, 0 );
    0-th cohomology functor of Category of left presentations of Z
    gap> cmor0 := ApplyFunctor( CH0, cocomplex_mor );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( cmor0 ) );
    2
    gap> CH1 := CohomologyFunctor( Zmod, 1 );
    1-th cohomology functor of Category of left presentations of Z
    gap> cmor1 := ApplyFunctor( CH1, cocomplex_mor );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( cmor1 ) );
    4
    gap> ToComplex := CocomplexToComplexFunctor( Zmod );
    Cocomplex to complex functor of Category of left presentations of Z
    gap> complex_mor := ApplyFunctor( ToComplex, cocomplex_mor );
    <A morphism in Complex category of Category of left presentations of Z>
    gap> H0 := HomologyFunctor( Zmod, 0 );
    0-th homology functor of Category of left presentations of Z
    gap> mor0 := ApplyFunctor( H0, complex_mor );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( mor0 ) );
    2
    gap> Hm1 := HomologyFunctor( Zmod, -1 );
    -1-th homology functor of Category of left presentations of Z
    gap> mor1 := ApplyFunctor( Hm1, complex_mor );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( mor1 ) );
    4
  
  
    Example  
    gap> QQ := HomalgFieldOfRationalsInSingular( );;
    gap> R := QQ * "x,y";
    Q[x,y]
    gap> SetRecursionTrapInterval( 10000 );
    gap> category := LeftPresentations( R );
    Category of left presentations of Q[x,y]
    gap> S := FreeLeftPresentation( 1, R );
    <An object in Category of left presentations of Q[x,y]>
    gap> object_func := function( i ) return S; end;
    function( i ) ... end
    gap> morphism_func := function( i ) return IdentityMorphism( S ); end;
    function( i ) ... end
    gap> C0 := ZFunctorObjectExtendedByInitialAndIdentity( object_func, morphism_func, category, 0, 4 );
    <An object in Functors from integers into Category of left presentations of Q[x,y]>
    gap> S2 := FreeLeftPresentation( 2, R );
    <An object in Category of left presentations of Q[x,y]>
    gap> C1 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S2, S ], 1 ) ], 2 );
    <An object in Functors from integers into Category of left presentations of Q[x,y]>
    gap> C1 := ZFunctorObjectExtendedByInitialAndIdentity( C1, 2, 3 );
    <An object in Functors from integers into Category of left presentations of Q[x,y]>
    gap> C2 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S, S ], 1 ) ], 3 );
    <An object in Functors from integers into Category of left presentations of Q[x,y]>
    gap> C2 := ZFunctorObjectExtendedByInitialAndIdentity( C2, 3, 4 );
    <An object in Functors from integers into Category of left presentations of Q[x,y]>
    gap> delta_1_3 := PresentationMorphism( C1[3], HomalgMatrix( [ [ "x^2" ], [ "xy" ], [ "y^3"] ], 3, 1, R ), C0[3] );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> delta_1_2 := PresentationMorphism( C1[2], HomalgMatrix( [ [ "x^2" ], [ "xy" ] ], 2, 1, R ), C0[2] );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> delta1 := ZFunctorMorphism( C1, [ UniversalMorphismFromInitialObject( C0[1] ), UniversalMorphismFromInitialObject( C0[1] ), delta_1_2, delta_1_3 ], 0, C0 );
    <A morphism in Functors from integers into Category of left presentations of Q[x,y]>
    gap> delta1 := ZFunctorMorphismExtendedByInitialAndIdentity( delta1, 0, 3 );
    <A morphism in Functors from integers into Category of left presentations of Q[x,y]>
    gap> delta1 := AsAscendingFilteredMorphism( delta1 );
    <A morphism in Ascending filtered object category of Category of left presentations of Q[x,y]>
    gap> delta_2_3 := PresentationMorphism( C2[3], HomalgMatrix( [ [ "y", "-x", "0" ] ], 1, 3, R ), C1[3] );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> delta_2_4 := PresentationMorphism( C2[4], HomalgMatrix( [ [ "y", "-x", "0" ], [ "0", "y^2", "-x" ] ], 2, 3, R ), C1[4] );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> delta2 := ZFunctorMorphism( C2, [  UniversalMorphismFromInitialObject( C1[2] ), delta_2_3, delta_2_4 ], 2, C1 );
    <A morphism in Functors from integers into Category of left presentations of Q[x,y]>
    gap> delta2 := ZFunctorMorphismExtendedByInitialAndIdentity( delta2, 2, 4 );
    <A morphism in Functors from integers into Category of left presentations of Q[x,y]>
    gap> delta2 := AsAscendingFilteredMorphism( delta2 );
    <A morphism in Ascending filtered object category of Category of left presentations of Q[x,y]>
    gap> SetIsAdditiveCategory( CategoryOfAscendingFilteredObjects( category ), true );
    gap> complex := ZFunctorObjectFromMorphismList( [ delta2, delta1 ], -2 );
    <An object in Functors from integers into Ascending filtered object category of Category of left presentations of Q[x,y]>
    gap> complex := AsComplex( complex );
    <An object in Complex category of Ascending filtered object category of Category of left presentations of Q[x,y]>
    gap> LessGenFunctor := FunctorLessGeneratorsLeft( R );
    Less generators for Category of left presentations of Q[x,y]
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 0, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 1 matrix)
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 1, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 1 matrix)
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 2, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 1 matrix)
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 3, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    x*y,
    x^2
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 4, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    x*y,
    x^2,
    y^3
    gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 5, 0, 0 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    x*y,
    x^2,
    y^3
    gap> s := SpectralSequenceDifferentialOfAscendingFilteredComplex( complex, 3, 3, -2 );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, s ) ) );
    y^3
    gap> AscToDescFunctor := AscendingToDescendingFilteredObjectFunctor( category );
    Ascending to descending filtered object functor of Category of left presentations of Q[x,y]
    gap> cocomplex := ZFunctorObjectFromMorphismList( [ ApplyFunctor( AscToDescFunctor, delta2 ), ApplyFunctor( AscToDescFunctor, delta1 ) ], -2 );
    <An object in Functors from integers into Descending filtered object category of Category of left presentations of Q[x,y]>
    gap> SetIsAdditiveCategory( CategoryOfDescendingFilteredObjects( category ), true );
    gap> cocomplex := AsCocomplex( cocomplex );
    <An object in Cocomplex category of Descending filtered object category of Category of left presentations of Q[x,y]>
    gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 0, -2, 1 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 2 matrix)
    gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 1, -2, 1 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 2 matrix)
    gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 2, -2, 1 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    -y,x
    gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 3, -2, 1 );
    <A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
    (an empty 0 x 0 matrix)
    gap> s := SpectralSequenceDifferentialOfDescendingFilteredCocomplex( cocomplex, 2, -2, 1 );
    <A morphism in Category of left presentations of Q[x,y]>
    gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, s ) ) );
    x^2,
    x*y
  
  
  
  12.2 Monoidal Categories
  
    Example  
    gap> ZZ := HomalgRingOfIntegers();;
    gap> Ml := AsLeftPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, ZZ ) );
    <An object in Category of left presentations of Z>
    gap> Nl := AsLeftPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZ ) );
    <An object in Category of left presentations of Z>
    gap> Tl := TensorProductOnObjects( Ml, Nl );
    <An object in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( Tl ) );
    [ [  3 ],
      [  2 ] ]
    gap> IsZeroForObjects( Tl );
    true
    gap> Bl := Braiding( DirectSum( Ml, Nl ), DirectSum( Ml, Ml ) );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( Bl ) );
    [ [  1,  0,  0,  0 ],
      [  0,  0,  1,  0 ],
      [  0,  1,  0,  0 ],
      [  0,  0,  0,  1 ] ]
    gap> IsWellDefined( Bl );
    true
    gap> Ul := TensorUnit( CapCategory( Ml ) );
    <An object in Category of left presentations of Z>
    gap> IntHoml := InternalHomOnObjects( DirectSum( Ml, Ul ), Nl );
    <An object in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( IntHoml ) );
    [ [  -2,  -1 ],
      [   1,  -1 ] ]
    gap> generator_l1 := StandardGeneratorMorphism( IntHoml, 1 );
    <A morphism in Category of left presentations of Z>
    gap> morphism_l1 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_l1 );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( morphism_l1 ) );
    [ [  0 ],
      [  2 ] ]
    gap> generator_l2 := StandardGeneratorMorphism( IntHoml, 2 );
    <A morphism in Category of left presentations of Z>
    gap> morphism_l2 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_l2 );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( morphism_l2 ) );
    [ [  0 ],
      [  2 ] ]
    gap> IsEqualForMorphisms( LambdaIntroduction( morphism_l1 ), generator_l1 );
    false
    gap> IsCongruentForMorphisms( LambdaIntroduction( morphism_l1 ), generator_l1 );
    true
    gap> IsEqualForMorphisms( LambdaIntroduction( morphism_l2 ), generator_l2 );
    false
    gap> IsCongruentForMorphisms( LambdaIntroduction( morphism_l2 ), generator_l2 );
    true
    gap> Mr := AsRightPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, ZZ ) );
    <An object in Category of right presentations of Z>
    gap> Nr := AsRightPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZ ) );
    <An object in Category of right presentations of Z>
    gap> Tr := TensorProductOnObjects( Mr, Nr );
    <An object in Category of right presentations of Z>
    gap> Display( UnderlyingMatrix( Tr ) );
    [ [  3,  2 ] ]
    gap> IsZeroForObjects( Tr );
    true
    gap> Br := Braiding( DirectSum( Mr, Nr ), DirectSum( Mr, Mr ) );
    <A morphism in Category of right presentations of Z>
    gap> Display( UnderlyingMatrix( Br ) );
    [ [  1,  0,  0,  0 ],
      [  0,  0,  1,  0 ],
      [  0,  1,  0,  0 ],
      [  0,  0,  0,  1 ] ]
    gap> IsWellDefined( Br );
    true
    gap> Ur := TensorUnit( CapCategory( Mr ) );
    <An object in Category of right presentations of Z>
    gap> IntHomr := InternalHomOnObjects( DirectSum( Mr, Ur ), Nr );
    <An object in Category of right presentations of Z>
    gap> Display( UnderlyingMatrix( IntHomr ) );
    [ [  -2,   1 ],
      [  -1,  -1 ] ]
    gap> generator_r1 := StandardGeneratorMorphism( IntHomr, 1 );
    <A morphism in Category of right presentations of Z>
    gap> morphism_r1 := LambdaElimination( DirectSum( Mr, Ur ), Nr, generator_r1 );
    <A morphism in Category of right presentations of Z>
    gap> Display( UnderlyingMatrix( morphism_r1 ) );
    [ [  0,   2 ] ]
    gap> generator_r2 := StandardGeneratorMorphism( IntHoml, 2 );
    <A morphism in Category of left presentations of Z>
    gap> morphism_r2 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_r2 );
    <A morphism in Category of left presentations of Z>
    gap> Display( UnderlyingMatrix( morphism_r2 ) );
    [ [   0 ],
      [   2 ] ]
    gap> IsEqualForMorphisms( LambdaIntroduction( morphism_r1 ), generator_r1 );
    false
    gap> IsCongruentForMorphisms( LambdaIntroduction( morphism_r1 ), generator_r1 );
    true
    gap> IsEqualForMorphisms( LambdaIntroduction( morphism_r2 ), generator_r2 );
    false
    gap> IsCongruentForMorphisms( LambdaIntroduction( morphism_r2 ), generator_r2 );
    true
  
  
  
  12.3 Generalized Morphisms Category
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForGeneralizedMorphismsTest" );
    VectorSpacesForGeneralizedMorphismsTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> LoadPackage( "GeneralizedMorphismsForCAP" );
    true
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> B_1 := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> C_1 := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> c1_source_aid := VectorSpaceMorphism( B_1, [ [ 1, 0 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ] ]
    
    gap> SetIsSubobject( c1_source_aid, true );
    gap> c1_range_aid := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ], C_1 );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ],
      [  0,  1 ],
      [  0,  0 ] ]
    
    gap> SetIsFactorobject( c1_range_aid, true );
    gap> c1_associated := VectorSpaceMorphism( B_1, [ [ 1, 1 ] ], C_1 );
    A rational vector space homomorphism with matrix: 
    [ [  1,  1 ] ]
    
    gap> c1 := GeneralizedMorphism( c1_source_aid, c1_associated, c1_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> B_2 := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> C_2 := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> c2_source_aid := VectorSpaceMorphism( B_2, [ [ 2, 0 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  2,  0 ] ]
    
    gap> SetIsSubobject( c2_source_aid, true );
    gap> c2_range_aid := VectorSpaceMorphism( C, [ [ 3, 0 ], [ 0, 3 ], [ 0, 0 ] ], C_2 );
    A rational vector space homomorphism with matrix: 
    [ [  3,  0 ],
      [  0,  3 ],
      [  0,  0 ] ]
    
    gap> SetIsFactorobject( c2_range_aid, true );
    gap> c2_associated := VectorSpaceMorphism( B_2, [ [ 6, 6 ] ], C_2 );
    A rational vector space homomorphism with matrix: 
    [ [  6,  6 ] ]
    
    gap> c2 := GeneralizedMorphism( c2_source_aid, c2_associated, c2_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> IsCongruentForMorphisms( c1, c2 );
    true
    gap> IsCongruentForMorphisms( c1, c1 );
    true
    gap> c3_associated := VectorSpaceMorphism( B_1, [ [ 2, 2 ] ], C_1 );
    A rational vector space homomorphism with matrix: 
    [ [  2,  2 ] ]
    
    gap> c3 := GeneralizedMorphism( c1_source_aid, c3_associated, c1_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> IsCongruentForMorphisms( c1, c3 );
    false
    gap> IsCongruentForMorphisms( c2, c3 );
    false
    gap> c1 + c2;
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> Arrow( c1 + c2 );
    A rational vector space homomorphism with matrix: 
    [ [  12,  12 ] ]
    
  
  
  First composition test:
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForGeneralizedMorphismsTest" );
    VectorSpacesForGeneralizedMorphismsTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> A := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> phi_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 2, 0 ] ], C );
    A rational vector space homomorphism with matrix: 
    [ [  1,  2,  0 ] ]
    
    gap> phi_tilde_source_aid := VectorSpaceMorphism( A, [ [ 1, 2 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  2 ] ]
    
    gap> phi_tilde := GeneralizedMorphismWithSourceAid( phi_tilde_source_aid, phi_tilde_associated );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> psi_tilde_associated := IdentityMorphism( B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ],
      [  0,  1 ] ]
    
    gap> psi_tilde_source_aid := VectorSpaceMorphism( B, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], C );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0,  0 ],
      [  0,  1,  0 ] ]
    
    gap> psi_tilde := GeneralizedMorphismWithSourceAid( psi_tilde_source_aid, psi_tilde_associated );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> composition := PreCompose( phi_tilde, psi_tilde );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> Arrow( composition );
    A rational vector space homomorphism with matrix: 
    [ [  1/2,    1 ] ]
    
    gap> SourceAid( composition );
    A rational vector space homomorphism with matrix: 
    [ [  1/2,    1 ] ]
    
    gap> RangeAid( composition );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ],
      [  0,  1 ] ]
  
  
  Second composition test
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForGeneralizedMorphismsTest" );
    VectorSpacesForGeneralizedMorphismsTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> A := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> phi2_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 5 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  5 ] ]
    
    gap> phi2_tilde_range_aid := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ],
      [  0,  1 ],
      [  1,  1 ] ]
    
    gap> phi2_tilde := GeneralizedMorphismWithRangeAid( phi2_tilde_associated, phi2_tilde_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> psi2_tilde_associated := VectorSpaceMorphism( C, [ [ 1 ], [ 3 ], [ 4 ] ], A );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  3 ],
      [  4 ] ]
    
    gap> psi2_tilde_range_aid := VectorSpaceMorphism( B, [ [ 1 ], [ 1 ] ], A );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ] ]
    
    gap> psi2_tilde := GeneralizedMorphismWithRangeAid( psi2_tilde_associated, psi2_tilde_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> composition2 := PreCompose( phi2_tilde, psi2_tilde );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> Arrow( composition2 );
    A rational vector space homomorphism with matrix: 
    [ [  16 ] ]
    
    gap> RangeAid( composition2 );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ] ]
    
    gap> SourceAid( composition2 );
    A rational vector space homomorphism with matrix: 
    [ [  1 ] ]
  
  
  Third composition test
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForGeneralizedMorphismsTest" );
    VectorSpacesForGeneralizedMorphismsTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> A := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Asub := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> B := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Bfac := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> Bsub := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Cfac := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> Asub_into_A := VectorSpaceMorphism( Asub, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], A );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0,  0 ],
      [  0,  1,  0 ] ]
    
    gap> Asub_to_Bfac := VectorSpaceMorphism( Asub, [ [ 1 ], [ 1 ] ], Bfac );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ] ]
    
    gap> B_onto_Bfac := VectorSpaceMorphism( B, [ [ 1 ], [ 1 ], [ 1 ] ], Bfac );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ],
      [  1 ] ]
    
    gap> Bsub_into_B := VectorSpaceMorphism( Bsub, [ [ 2, 2, 0 ], [ 0, 2, 2 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  2,  2,  0 ],
      [  0,  2,  2 ] ]
    
    gap> Bsub_to_Cfac := VectorSpaceMorphism( Bsub, [ [ 3 ], [ 0 ] ], Cfac );
    A rational vector space homomorphism with matrix: 
    [ [  3 ],
      [  0 ] ]
    
    gap> C_onto_Cfac := VectorSpaceMorphism( C, [ [ 1 ], [ 2 ], [ 3 ] ], Cfac );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  2 ],
      [  3 ] ]
    
    gap> generalized_morphism1 := GeneralizedMorphism( Asub_into_A, Asub_to_Bfac, B_onto_Bfac );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> generalized_morphism2 := GeneralizedMorphism( Bsub_into_B, Bsub_to_Cfac, C_onto_Cfac );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> IsWellDefined( generalized_morphism1 );
    true
    gap> IsWellDefined( generalized_morphism2 );
    true
    gap> p := PreCompose( generalized_morphism1, generalized_morphism2 );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> SourceAid( p );
    A rational vector space homomorphism with matrix: 
    [ [  -1,   1,   0 ],
      [   1,   0,   0 ] ]
    
    gap> Arrow( p );
    A rational vector space homomorphism with matrix: 
    (an empty 2 x 0 matrix)
    
    gap> RangeAid( p );
    A rational vector space homomorphism with matrix: 
    (an empty 3 x 0 matrix)
    gap> A := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Asub := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> B := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Bfac := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> Bsub := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> Cfac := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> Asub_into_A := VectorSpaceMorphism( Asub, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], A );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0,  0 ],
      [  0,  1,  0 ] ]
    
    gap> Asub_to_Bfac := VectorSpaceMorphism( Asub, [ [ 1 ], [ 1 ] ], Bfac );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ] ]
    
    gap> B_onto_Bfac := VectorSpaceMorphism( B, [ [ 1 ], [ 1 ], [ 1 ] ], Bfac );
    A rational vector space homomorphism with matrix: 
    [ [  1 ],
      [  1 ],
      [  1 ] ]
    
    gap> Bsub_into_B := VectorSpaceMorphism( Bsub, [ [ 2, 2, 0 ], [ 0, 2, 2 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  2,  2,  0 ],
      [  0,  2,  2 ] ]
    
    gap> Bsub_to_Cfac := VectorSpaceMorphism( Bsub, [ [ 3, 3 ], [ 0, 0 ] ], Cfac );
    A rational vector space homomorphism with matrix: 
    [ [  3,  3 ],
      [  0,  0 ] ]
    
    gap> C_onto_Cfac := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 2 ], [ 3, 3 ] ], Cfac );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0 ],
      [  0,  2 ],
      [  3,  3 ] ]
    
    gap> generalized_morphism1 := GeneralizedMorphism( Asub_into_A, Asub_to_Bfac, B_onto_Bfac );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> generalized_morphism2 := GeneralizedMorphism( Bsub_into_B, Bsub_to_Cfac, C_onto_Cfac );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> IsWellDefined( generalized_morphism1 );
    true
    gap> IsWellDefined( generalized_morphism2 );
    true
    gap> p := PreCompose( generalized_morphism1, generalized_morphism2 );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> SourceAid( p );
    A rational vector space homomorphism with matrix: 
    [ [  -1,   1,   0 ],
      [   1,   0,   0 ] ]
    
    gap> Arrow( p );
    A rational vector space homomorphism with matrix: 
    [ [  0 ],
      [  0 ] ]
    
    gap> RangeAid( p );
    A rational vector space homomorphism with matrix: 
    [ [  -1 ],
      [   2 ],
      [   0 ] ]
  
  
  Honest representative test
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForGeneralizedMorphismsTest" );
    VectorSpacesForGeneralizedMorphismsTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> A := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> phi_tilde_source_aid := VectorSpaceMorphism( A, [ [ 2 ] ], A );
    A rational vector space homomorphism with matrix: 
    [ [  2 ] ]
    
    gap> phi_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 1 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  1 ] ]
    
    gap> phi_tilde_range_aid := VectorSpaceMorphism( B, [ [ 1, 2 ], [ 3, 4 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  2 ],
      [  3,  4 ] ]
    
    gap> phi_tilde := GeneralizedMorphism( phi_tilde_source_aid, phi_tilde_associated, phi_tilde_range_aid );
    <A morphism in Generalized morphism category of VectorSpacesForGeneralizedMorphismsTest>
    gap> HonestRepresentative( phi_tilde );
    A rational vector space homomorphism with matrix: 
    [ [  -1/4,   1/4 ] ]
    
    gap> IsWellDefined( phi_tilde );
    true
    gap> IsWellDefined( psi_tilde );
    true
  
  
  
  12.4 IsWellDefined
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForIsWellDefinedTest" );
    VectorSpacesForIsWellDefinedTest 
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> LoadPackage( "GeneralizedMorphismsForCAP" );
    true
    gap> A := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> alpha := VectorSpaceMorphism( A, [ [ 1, 2 ] ], B );
    A rational vector space homomorphism with matrix: 
    [ [  1,  2 ] ]
    
    gap> g := GeneralizedMorphism( alpha, alpha, alpha );
    <A morphism in Generalized morphism category of VectorSpacesForIsWellDefinedTest>
    gap> IsWellDefined( alpha );
    true
    gap> IsWellDefined( g );
    true
  
  
  
  12.5 Kernel
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpaces01" );
    VectorSpaces01
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAddKernel01.gi" );
    true
    gap> V := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> W := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W );
    A rational vector space homomorphism with matrix: 
    [ [   1,   1,   1 ],
      [  -1,  -1,  -1 ] ]
    
    gap> k := KernelObject( alpha );
    <A rational vector space of dimension 1>
    gap> T := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> tau := VectorSpaceMorphism( T, [ [ 2, 2 ], [ 2, 2 ] ], V );
    A rational vector space homomorphism with matrix: 
    [ [  2,  2 ],
      [  2,  2 ] ]
    
    gap> k_lift := KernelLift( alpha, tau );
    A rational vector space homomorphism with matrix: 
    [ [  2 ],
      [  2 ] ]
    
    gap> HasKernelEmbedding( alpha );
    false
    gap> KernelEmbedding( alpha );
    A rational vector space homomorphism with matrix: 
    [ [  1,  1 ] ]
    
  
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpaces02" );
    VectorSpaces02
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAddKernel02.gi" );
    true
    gap> V := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> W := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W );
    A rational vector space homomorphism with matrix: 
    [ [   1,   1,   1 ],
      [  -1,  -1,  -1 ] ]
    
    gap> k := KernelObject( alpha );
    <A rational vector space of dimension 1>
    gap> T := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> tau := VectorSpaceMorphism( T, [ [ 2, 2 ], [ 2, 2 ] ], V );
    A rational vector space homomorphism with matrix: 
    [ [  2,  2 ],
      [  2,  2 ] ]
    
    gap> k_lift := KernelLift( alpha, tau );
    A rational vector space homomorphism with matrix: 
    [ [  2 ],
      [  2 ] ]
    
    gap> HasKernelEmbedding( alpha );
    false
  
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpaces03" );
    VectorSpaces03
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAddKernel03.gi" );
    true
    gap> V := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> W := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W );
    A rational vector space homomorphism with matrix: 
    [ [   1,   1,   1 ],
      [  -1,  -1,  -1 ] ]
    
    gap> k := KernelObject( alpha );
    <A rational vector space of dimension 1>
    gap> k_emb := KernelEmbedding( alpha );
    A rational vector space homomorphism with matrix: 
    [ [  1,  1 ] ]
    
    gap> IsIdenticalObj( Source( k_emb ), k );
    true
    gap> V := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> W := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> beta := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W );
    A rational vector space homomorphism with matrix: 
    [ [   1,   1,   1 ],
      [  -1,  -1,  -1 ] ]
    
    gap> k_emb := KernelEmbedding( beta );
    A rational vector space homomorphism with matrix: 
    [ [  1,  1 ] ]
    
    gap> IsIdenticalObj( Source( k_emb ), KernelObject( beta ) );
    true
  
  
  
  12.6 FiberProduct
  
    Example  
    gap> vecspaces := CreateCapCategory( "VectorSpacesForFiberProductTest" );
    VectorSpacesForFiberProductTest
    gap> ReadPackage( "CAP", "examples/testfiles/VectorSpacesAllMethods.gi" );
    true
    gap> A := QVectorSpace( 1 );
    <A rational vector space of dimension 1>
    gap> B := QVectorSpace( 2 );
    <A rational vector space of dimension 2>
    gap> C := QVectorSpace( 3 );
    <A rational vector space of dimension 3>
    gap> AtoC := VectorSpaceMorphism( A, [ [ 1, 2, 0 ] ], C );
    A rational vector space homomorphism with matrix: 
    [ [  1,  2,  0 ] ]
    
    gap> BtoC := VectorSpaceMorphism( B, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], C );
    A rational vector space homomorphism with matrix: 
    [ [  1,  0,  0 ],
      [  0,  1,  0 ] ]
    
    gap> P := FiberProduct( AtoC, BtoC );
    <A rational vector space of dimension 1>
    gap> p1 := ProjectionInFactorOfFiberProduct( [ AtoC, BtoC ], 1 );
    A rational vector space homomorphism with matrix: 
    [ [  1/2 ] ]
    
    gap> p2 := ProjectionInFactorOfFiberProduct( [ AtoC, BtoC ], 2 );
    A rational vector space homomorphism with matrix: 
    [ [  1/2,    1 ] ]