  
  [1X15 [33X[0;0YExamples and Tests[133X[101X
  
  
  [1X15.1 [33X[0;0YDummy implementations[133X[101X
  
  
  [1X15.1-1 [33X[0;0YDummy categories[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CAP", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xlist_of_operations_to_install := [[127X[104X
    [4X[25X>[125X [27X    "ObjectConstructor",[127X[104X
    [4X[25X>[125X [27X    "MorphismConstructor",[127X[104X
    [4X[25X>[125X [27X    "ObjectDatum",[127X[104X
    [4X[25X>[125X [27X    "MorphismDatum",[127X[104X
    [4X[25X>[125X [27X    "IsCongruentForMorphisms",[127X[104X
    [4X[25X>[125X [27X    "PreCompose",[127X[104X
    [4X[25X>[125X [27X    "IdentityMorphism",[127X[104X
    [4X[25X>[125X [27X    "DirectSum",[127X[104X
    [4X[25X>[125X [27X];;[127X[104X
    [4X[25Xgap>[125X [27Xdummy := DummyCategory( rec([127X[104X
    [4X[25X>[125X [27X    list_of_operations_to_install := list_of_operations_to_install,[127X[104X
    [4X[25X>[125X [27X    properties := [ "IsAdditiveCategory" ],[127X[104X
    [4X[25X>[125X [27X) );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( list_of_operations_to_install, o -> CanCompute( dummy, o ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAdditiveCategory( dummy );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.1-2 [33X[0;0YDummy rings[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CAP", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDummyRing( );[127X[104X
    [4X[28XDummy ring 1[128X[104X
    [4X[25Xgap>[125X [27XDummyRing( );[127X[104X
    [4X[28XDummy ring 2[128X[104X
    [4X[25Xgap>[125X [27XIsRing( DummyRing( ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDummyCommutativeRing( );[127X[104X
    [4X[28XDummy commutative ring 1[128X[104X
    [4X[25Xgap>[125X [27XDummyCommutativeRing( );[127X[104X
    [4X[28XDummy commutative ring 2[128X[104X
    [4X[25Xgap>[125X [27XIsRing( DummyCommutativeRing( ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCommutative( DummyCommutativeRing( ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDummyField( );[127X[104X
    [4X[28XDummy field 1[128X[104X
    [4X[25Xgap>[125X [27XDummyField( );[127X[104X
    [4X[28XDummy field 2[128X[104X
    [4X[25Xgap>[125X [27XIsRing( DummyField( ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsField( DummyField( ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.2 [33X[0;0YFunctors[133X[101X
  
  [33X[0;0YWe  create  a  binary  functor  [23XF[123X  with  one covariant and one contravariant
  component in two ways. Here is the first way to model a binary functor:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfield := HomalgFieldOfRationals( );;[127X[104X
    [4X[25Xgap>[125X [27Xvec := LeftPresentations( field );;[127X[104X
    [4X[25Xgap>[125X [27XF := CapFunctor( "CohomForVec", [ vec, [ vec, true ] ], vec );;[127X[104X
    [4X[25Xgap>[125X [27Xobj_func := function( A, B ) return TensorProductOnObjects( A, DualOnObjects( B ) ); end;;[127X[104X
    [4X[25Xgap>[125X [27Xmor_func := function( source, alpha, beta, range ) return TensorProductOnMorphismsWithGivenTensorProducts( source, alpha, DualOnMorphisms( beta ), range ); end;;[127X[104X
    [4X[25Xgap>[125X [27XAddObjectFunction( F, obj_func );;[127X[104X
    [4X[25Xgap>[125X [27XAddMorphismFunction( F, mor_func );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YCAP  regards  [23XF[123X  as  a binary functor on a technical level, as we can see by
  looking at its input signature:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInputSignature( F );[127X[104X
    [4X[28X[ [ Category of left presentations of Q, false ], [ Category of left presentations of Q, true ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe can see that [10XApplyFunctor[110X works both on two arguments and on one argument
  (in the product category).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV1 := TensorUnit( vec );;[127X[104X
    [4X[25Xgap>[125X [27XV3 := DirectSum( V1, V1, V1 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1 := ProjectionInFactorOfDirectSum( [ V1, V1 ], 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2 := ProjectionInFactorOfDirectSum( [ V3, V1 ], 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xvalue1 := ApplyFunctor( F, pi1, pi2 );;[127X[104X
    [4X[25Xgap>[125X [27Xinput := Product( pi1, Opposite( pi2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xvalue2 := ApplyFunctor( F, input );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( value1, value2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere is the second way to model a binary functor:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF2 := CapFunctor( "CohomForVec2", Product( vec, Opposite( vec ) ), vec );;[127X[104X
    [4X[25Xgap>[125X [27XAddObjectFunction( F2, a -> obj_func( a[1], Opposite( a[2] ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XAddMorphismFunction( F2, function( source, datum, range ) return mor_func( source, datum[1], Opposite( datum[2] ), range ); end );;[127X[104X
    [4X[25Xgap>[125X [27Xvalue3 := ApplyFunctor( F2,input );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( value1, value3 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YCAP  regards  [23XF2[123X  as  a unary functor on a technical level, as we can see by
  looking at its input signature:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInputSignature( F2 );[127X[104X
    [4X[28X[ [ Product of: Category of left presentations of Q, Opposite( Category of left presentations of Q ), false ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YInstallation  of  the first functor as a GAP-operation. It will be installed
  both as a unary and binary version.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallFunctor( F, "F_installation" );;[127X[104X
    [4X[25Xgap>[125X [27XF_installation( pi1, pi2 );;[127X[104X
    [4X[25Xgap>[125X [27XF_installation( input );;[127X[104X
    [4X[25Xgap>[125X [27XF_installationOnObjects( V1, V1 );;[127X[104X
    [4X[25Xgap>[125X [27XF_installationOnObjects( Product( V1, Opposite( V1 ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XF_installationOnMorphisms( pi1, pi2 );;[127X[104X
    [4X[25Xgap>[125X [27XF_installationOnMorphisms( input );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YInstallation  of the second functor as a GAP-operation. It will be installed
  only as a unary version.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallFunctor( F2, "F_installation2" );;[127X[104X
    [4X[25Xgap>[125X [27XF_installation2( input );;[127X[104X
    [4X[25Xgap>[125X [27XF_installation2OnObjects( Product( V1, Opposite( V1 ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XF_installation2OnMorphisms( input );;[127X[104X
  [4X[32X[104X
  
  
  [1X15.3 [33X[0;0YHandlePrecompiledTowers[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CAP", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xdummy1 := CreateCapCategory( );;[127X[104X
    [4X[25Xgap>[125X [27Xdummy2 := CreateCapCategory( );;[127X[104X
    [4X[25Xgap>[125X [27Xdummy3 := CreateCapCategory( );;[127X[104X
    [4X[25Xgap>[125X [27XPrintAndReturn := function ( string )[127X[104X
    [4X[25X>[125X [27X    Print( string, "\n" ); return string; end;;[127X[104X
    [4X[25Xgap>[125X [27Xdummy1!.compiler_hints := rec( );;[127X[104X
    [4X[25Xgap>[125X [27Xdummy1!.compiler_hints.precompiled_towers := [[127X[104X
    [4X[25X>[125X [27X  rec([127X[104X
    [4X[25X>[125X [27X    remaining_constructors_in_tower := [ "Constructor1" ],[127X[104X
    [4X[25X>[125X [27X    precompiled_functions_adder := cat ->[127X[104X
    [4X[25X>[125X [27X      PrintAndReturn( "Adding precompiled operations for Constructor1" ),[127X[104X
    [4X[25X>[125X [27X  ),[127X[104X
    [4X[25X>[125X [27X  rec([127X[104X
    [4X[25X>[125X [27X    remaining_constructors_in_tower := [ "Constructor1", "Constructor2" ],[127X[104X
    [4X[25X>[125X [27X    precompiled_functions_adder := cat ->[127X[104X
    [4X[25X>[125X [27X      PrintAndReturn( "Adding precompiled operations for Constructor2" ),[127X[104X
    [4X[25X>[125X [27X  ),[127X[104X
    [4X[25X>[125X [27X];;[127X[104X
    [4X[25Xgap>[125X [27XHandlePrecompiledTowers( dummy2, dummy1, "Constructor1" );[127X[104X
    [4X[28XAdding precompiled operations for Constructor1[128X[104X
    [4X[25Xgap>[125X [27XHandlePrecompiledTowers( dummy3, dummy2, "Constructor2" );[127X[104X
    [4X[28XAdding precompiled operations for Constructor2[128X[104X
  [4X[32X[104X
  
  
  [1X15.4 [33X[0;0YHomomorphism structure[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XReadPackage( "CAP", "examples/FieldAsCategory.g" );;[127X[104X
    [4X[25Xgap>[125X [27XQ := HomalgFieldOfRationals();;[127X[104X
    [4X[25Xgap>[125X [27XQoid := FieldAsCategory( Q );;[127X[104X
    [4X[25Xgap>[125X [27Xa := FieldAsCategoryMorphism( Qoid, 1/2 );;[127X[104X
    [4X[25Xgap>[125X [27Xb := FieldAsCategoryMorphism( Qoid, -2/3 );;[127X[104X
    [4X[25Xgap>[125X [27Xu := FieldAsCategoryUniqueObject( Qoid );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( a,[127X[104X
    [4X[25X>[125X [27X    InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism([127X[104X
    [4X[25X>[125X [27X        u,u,[127X[104X
    [4X[25X>[125X [27X        InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure([127X[104X
    [4X[25X>[125X [27X            a[127X[104X
    [4X[25X>[125X [27X        )[127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xa = HomStructure( u, u, HomStructure( a ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForObjects( HomStructure( Qoid ), DistinguishedObjectOfHomomorphismStructure( Qoid ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xc := FieldAsCategoryMorphism( Qoid, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xd := FieldAsCategoryMorphism( Qoid, 0 );;[127X[104X
    [4X[25Xgap>[125X [27Xleft_coeffs := [ [ a, b ], [ c, d ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xright_coeffs := [ [ PreCompose( a, b ), PreCompose( b, c ) ], [ c, PreCompose( a, a ) ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xright_side := [ a, b ];;[127X[104X
    [4X[25Xgap>[125X [27XMereExistenceOfSolutionOfLinearSystemInAbCategory( left_coeffs, right_coeffs, right_side );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xsolution := [127X[104X
    [4X[25X>[125X [27X    SolveLinearSystemInAbCategory([127X[104X
    [4X[25X>[125X [27X    left_coeffs,[127X[104X
    [4X[25X>[125X [27X    right_coeffs,[127X[104X
    [4X[25X>[125X [27X    right_side[127X[104X
    [4X[25X>[125X [27X);;[127X[104X
    [4X[25Xgap>[125X [27XForAll( [ 1, 2 ], i ->[127X[104X
    [4X[25X>[125X [27X    IsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X        Sum( List( [ 1, 2 ], j -> PreCompose( [ left_coeffs[i][j], solution[j], right_coeffs[i][j] ] ) ) ),[127X[104X
    [4X[25X>[125X [27X        right_side[i][127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLiftable( c, d );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XLiftOrFail( c, d );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XIsLiftable( d, c );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLiftOrFail( d, c );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLift( d, c );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XIsColiftable( c, d );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XColiftOrFail( c, d );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XColift( c, d );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XIsColiftable( d, c );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XColiftOrFail( d, c );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XReadPackage( "CAP", "examples/StringsAsCategory.g" );;[127X[104X
    [4X[25Xgap>[125X [27XC := StringsAsCategory();;[127X[104X
    [4X[25Xgap>[125X [27Xobj1 := StringsAsCategoryObject( C, "qaeiou" );;[127X[104X
    [4X[25Xgap>[125X [27Xobj2 := StringsAsCategoryObject( C, "qxayeziouT" );;[127X[104X
    [4X[25Xgap>[125X [27Xmor := StringsAsCategoryMorphism( C, obj1, "xyzaTe", obj2 );;[127X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifyObject[127X[104X
    [4X[25X>[125X [27XIsEqualForObjects( SimplifyObject( obj1, 0 ), obj1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForObjects( SimplifyObject( obj1, 1 ), obj1 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1,2,3,4], i -> IsEqualForObjects( SimplifyObject( obj1, i ), SimplifyObject( obj1, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [5,6,7,8], i -> IsEqualForObjects( SimplifyObject( obj1, i ), SimplifyObject( obj1, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifyMorphism[127X[104X
    [4X[25X>[125X [27XIsEqualForMorphisms( SimplifyMorphism( mor, 0 ), mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( SimplifyMorphism( mor, 1 ), mor );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1], i -> IsEqualForMorphisms( SimplifyMorphism( mor, i ), SimplifyMorphism( mor, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [2,3,4,5], i -> IsEqualForMorphisms( SimplifyMorphism( mor, i ), SimplifyMorphism( mor, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifySource[127X[104X
    [4X[25X>[125X [27XIsEqualForMorphismsOnMor( SimplifySource( mor, 0 ), mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphismsOnMor( SimplifySource( mor, 1 ), mor );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySource( mor, i ), SimplifySource( mor, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySource( mor, i ), SimplifySource( mor, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    PreCompose( SimplifySource_IsoFromInputObject( mor, infinity ), SimplifySource( mor, infinity ) ), mor[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    PreCompose( SimplifySource_IsoToInputObject( mor, infinity ), mor ) , SimplifySource( mor, infinity )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifyRange[127X[104X
    [4X[25X>[125X [27XIsEqualForMorphismsOnMor( SimplifyRange( mor, 0 ), mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphismsOnMor( SimplifyRange( mor, 1 ), mor );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifyRange( mor, i ), SimplifyRange( mor, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifyRange( mor, i ), SimplifyRange( mor, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    PreCompose( SimplifyRange( mor, infinity ), SimplifyRange_IsoToInputObject( mor, infinity ) ), mor[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    PreCompose( mor, SimplifyRange_IsoFromInputObject( mor, infinity ) ), SimplifyRange( mor, infinity )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifySourceAndRange[127X[104X
    [4X[25X>[125X [27XIsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, 0 ), mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, 1 ), mor );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, i ), SimplifySourceAndRange( mor, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, i ), SimplifySourceAndRange( mor, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    mor,[127X[104X
    [4X[25X>[125X [27X    PreCompose( [ SimplifySourceAndRange_IsoFromInputSource( mor, infinity ),[127X[104X
    [4X[25X>[125X [27X                  SimplifySourceAndRange( mor, infinity ),[127X[104X
    [4X[25X>[125X [27X                  SimplifySourceAndRange_IsoToInputRange( mor, infinity ) ] )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    SimplifySourceAndRange( mor, infinity ),[127X[104X
    [4X[25X>[125X [27X    PreCompose( [ SimplifySourceAndRange_IsoToInputSource( mor, infinity ),[127X[104X
    [4X[25X>[125X [27X                  mor,[127X[104X
    [4X[25X>[125X [27X                  SimplifySourceAndRange_IsoFromInputRange( mor, infinity ) ] )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X## Test SimplifyEndo[127X[104X
    [4X[25X>[125X [27Xendo := StringsAsCategoryMorphism( C, obj1, "uoiea", obj1 );;[127X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( endo );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphismsOnMor( SimplifyEndo( endo, 0 ), endo );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphismsOnMor( SimplifyEndo( endo, 1 ), endo );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( endo, i ), SimplifySourceAndRange( endo, i + 1 ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( endo, i ), SimplifySourceAndRange( endo, i + 1 ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xiota := SimplifyEndo_IsoToInputObject( endo, infinity );;[127X[104X
    [4X[25Xgap>[125X [27Xiota_inv := SimplifyEndo_IsoFromInputObject( endo, infinity );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PreCompose( [ iota_inv, SimplifyEndo( endo, infinity ), iota ] ), endo );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.5 [33X[0;0YHomology object[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfield := HomalgFieldOfRationals( );;[127X[104X
    [4X[25Xgap>[125X [27XA := VectorSpaceObject( 1, field );;[127X[104X
    [4X[25Xgap>[125X [27XB := VectorSpaceObject( 2, field );;[127X[104X
    [4X[25Xgap>[125X [27XC := VectorSpaceObject( 3, field );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha := VectorSpaceMorphism( A, HomalgMatrix( [ [ 1, 0, 0 ] ], 1, 3, field ), C );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta := VectorSpaceMorphism( C, HomalgMatrix( [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ], 3, 2, field ), B );;[127X[104X
    [4X[25Xgap>[125X [27XIsZero( PreCompose( alpha, beta ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    IdentityMorphism( HomologyObject( alpha, beta ) ),[127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( alpha, beta, IdentityMorphism( C ), alpha, beta )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xkernel_beta := KernelEmbedding( beta );;[127X[104X
    [4X[25Xgap>[125X [27XK := Source( kernel_beta );;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism([127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( [127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( K ), [127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( K ),[127X[104X
    [4X[25X>[125X [27X        kernel_beta,[127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( Source( beta ) ),[127X[104X
    [4X[25X>[125X [27X        beta[127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcokernel_alpha := CokernelProjection( alpha );;[127X[104X
    [4X[25Xgap>[125X [27XCo := Range( cokernel_alpha );;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism([127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( [127X[104X
    [4X[25X>[125X [27X        alpha,[127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( Range( alpha ) ),[127X[104X
    [4X[25X>[125X [27X        cokernel_alpha,[127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( Co ),[127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( Co )[127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xalpha_op := Opposite( alpha );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta_op := Opposite( beta );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    IdentityMorphism( HomologyObject( beta_op, alpha_op ) ),[127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( beta_op, alpha_op, IdentityMorphism( Opposite( C ) ), beta_op, alpha_op )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xkernel_beta := KernelEmbedding( beta_op );;[127X[104X
    [4X[25Xgap>[125X [27XK := Source( kernel_beta );;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism([127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( [127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( K ), [127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( K ),[127X[104X
    [4X[25X>[125X [27X        kernel_beta,[127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( Source( beta_op ) ),[127X[104X
    [4X[25X>[125X [27X        beta_op[127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcokernel_alpha := CokernelProjection( alpha_op );;[127X[104X
    [4X[25Xgap>[125X [27XCo := Range( cokernel_alpha );;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism([127X[104X
    [4X[25X>[125X [27X    HomologyObjectFunctorial( [127X[104X
    [4X[25X>[125X [27X        alpha_op,[127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( Range( alpha_op ) ),[127X[104X
    [4X[25X>[125X [27X        cokernel_alpha,[127X[104X
    [4X[25X>[125X [27X        MorphismFromZeroObject( Co ),[127X[104X
    [4X[25X>[125X [27X        MorphismIntoZeroObject( Co )[127X[104X
    [4X[25X>[125X [27X    )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.6 [33X[0;0YLiftable[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfield := HomalgFieldOfRationals( );;[127X[104X
    [4X[25Xgap>[125X [27XV := VectorSpaceObject( 1, field );;[127X[104X
    [4X[25Xgap>[125X [27XW := VectorSpaceObject( 2, field );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha := VectorSpaceMorphism( V, HomalgMatrix( [ [ 1, -1 ] ], 1, 2, field ), W );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta := VectorSpaceMorphism( W, HomalgMatrix( [ [ 1, 2 ], [ 3, 4 ] ], 2, 2, field ), W );;[127X[104X
    [4X[25Xgap>[125X [27XIsLiftable( alpha, beta );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLiftable( beta, alpha );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsLiftableAlongMonomorphism( beta, alpha );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgamma := VectorSpaceMorphism( W, HomalgMatrix( [ [ 1 ], [ 1 ] ], 2, 1, field ), V );;[127X[104X
    [4X[25Xgap>[125X [27XIsColiftable( beta, gamma );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsColiftable( gamma, beta );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsColiftableAlongEpimorphism( beta, gamma );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XPreCompose( PreInverseForMorphisms( gamma ), gamma ) = IdentityMorphism( V );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XPreCompose( alpha, PostInverseForMorphisms( alpha ) ) = IdentityMorphism( V );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.7 [33X[0;0YMonoidal Categories[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZZ := HomalgRingOfIntegers();;[127X[104X
    [4X[25Xgap>[125X [27XMl := AsLeftPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, ZZZ ) );[127X[104X
    [4X[28X<An object in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XNl := AsLeftPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZZ ) );[127X[104X
    [4X[28X<An object in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XTl := TensorProductOnObjects( Ml, Nl );[127X[104X
    [4X[28X<An object in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( Tl ) );[127X[104X
    [4X[28X[ [  3 ],[128X[104X
    [4X[28X  [  2 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsZeroForObjects( Tl );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XBl := Braiding( DirectSum( Ml, Nl ), DirectSum( Ml, Ml ) );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( Bl ) );[127X[104X
    [4X[28X[ [  1,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1,  0 ],[128X[104X
    [4X[28X  [  0,  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( Bl );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUl := TensorUnit( CapCategory( Ml ) );[127X[104X
    [4X[28X<An object in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XIntHoml := InternalHomOnObjects( DirectSum( Ml, Ul ), Nl );[127X[104X
    [4X[28X<An object in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( IntHoml ) );[127X[104X
    [4X[28X[ [  1,  2 ],[128X[104X
    [4X[28X  [  0,  3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgenerator_l1 := StandardGeneratorMorphism( IntHoml, 1 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27Xmorphism_l1 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_l1 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( morphism_l1 ) );[127X[104X
    [4X[28X[ [  -3 ],[128X[104X
    [4X[28X  [   2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgenerator_l2 := StandardGeneratorMorphism( IntHoml, 2 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27Xmorphism_l2 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_l2 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( morphism_l2 ) );[127X[104X
    [4X[28X[ [   0 ],[128X[104X
    [4X[28X  [  -1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( LambdaIntroduction( morphism_l1 ), generator_l1 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( LambdaIntroduction( morphism_l1 ), generator_l1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( LambdaIntroduction( morphism_l2 ), generator_l2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( LambdaIntroduction( morphism_l2 ), generator_l2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMr := AsRightPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, ZZZ ) );[127X[104X
    [4X[28X<An object in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XNr := AsRightPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZZ ) );[127X[104X
    [4X[28X<An object in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XTr := TensorProductOnObjects( Mr, Nr );[127X[104X
    [4X[28X<An object in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( Tr ) );[127X[104X
    [4X[28X[ [  3,  2 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsZeroForObjects( Tr );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XBr := Braiding( DirectSum( Mr, Nr ), DirectSum( Mr, Mr ) );[127X[104X
    [4X[28X<A morphism in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( Br ) );[127X[104X
    [4X[28X[ [  1,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1,  0 ],[128X[104X
    [4X[28X  [  0,  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( Br );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUr := TensorUnit( CapCategory( Mr ) );[127X[104X
    [4X[28X<An object in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XIntHomr := InternalHomOnObjects( DirectSum( Mr, Ur ), Nr );[127X[104X
    [4X[28X<An object in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( IntHomr ) );[127X[104X
    [4X[28X[ [  1,  0 ],[128X[104X
    [4X[28X  [  2,  3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgenerator_r1 := StandardGeneratorMorphism( IntHomr, 1 );[127X[104X
    [4X[28X<A morphism in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27Xmorphism_r1 := LambdaElimination( DirectSum( Mr, Ur ), Nr, generator_r1 );[127X[104X
    [4X[28X<A morphism in Category of right presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( morphism_r1 ) );[127X[104X
    [4X[28X[ [  -3,   2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgenerator_r2 := StandardGeneratorMorphism( IntHoml, 2 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27Xmorphism_r2 := LambdaElimination( DirectSum( Ml, Ul ), Nl, generator_r2 );[127X[104X
    [4X[28X<A morphism in Category of left presentations of Z>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( UnderlyingMatrix( morphism_r2 ) );[127X[104X
    [4X[28X[ [   0 ],[128X[104X
    [4X[28X  [  -1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( LambdaIntroduction( morphism_r1 ), generator_r1 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( LambdaIntroduction( morphism_r1 ), generator_r1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( LambdaIntroduction( morphism_r2 ), generator_r2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( LambdaIntroduction( morphism_r2 ), generator_r2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.8 [33X[0;0YMorphismFromSourceToPushout and MorphismFromFiberProductToSink[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfield := HomalgFieldOfRationals( );;[127X[104X
    [4X[25Xgap>[125X [27XA := VectorSpaceObject( 3, field );;[127X[104X
    [4X[25Xgap>[125X [27XB := VectorSpaceObject( 2, field );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha := VectorSpaceMorphism( B, HomalgMatrix( [ [ 1, -1, 1 ], [ 1, 1, 1 ] ], 2, 3, field ), A );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta := VectorSpaceMorphism( B, HomalgMatrix( [ [ 1, 2, 1 ], [ 2, 1, 1 ] ], 2, 3, field ), A );;[127X[104X
    [4X[25Xgap>[125X [27Xm := MorphismFromFiberProductToSink( [ alpha, beta ] );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    m,[127X[104X
    [4X[25X>[125X [27X    PreCompose( ProjectionInFactorOfFiberProduct( [ alpha, beta ], 1 ), alpha )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    m,[127X[104X
    [4X[25X>[125X [27X    PreCompose( ProjectionInFactorOfFiberProduct( [ alpha, beta ], 2 ), beta )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27XMorphismFromKernelObjectToSink( alpha ),[127X[104X
    [4X[25X>[125X [27X    PreCompose( KernelEmbedding( alpha ), alpha )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xalpha_p := DualOnMorphisms( alpha );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta_p := DualOnMorphisms( beta );;[127X[104X
    [4X[25Xgap>[125X [27Xm_p := MorphismFromSourceToPushout( [ alpha_p, beta_p ] );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    m_p,[127X[104X
    [4X[25X>[125X [27X    PreCompose( alpha_p, InjectionOfCofactorOfPushout( [ alpha_p, beta_p ], 1 ) )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    m_p,[127X[104X
    [4X[25X>[125X [27X    PreCompose( beta_p, InjectionOfCofactorOfPushout( [ alpha_p, beta_p ], 2 ) )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms([127X[104X
    [4X[25X>[125X [27X    MorphismFromSourceToCokernelObject( alpha_p ),[127X[104X
    [4X[25X>[125X [27X    PreCompose( alpha_p, CokernelProjection( alpha_p ) )[127X[104X
    [4X[25X>[125X [27X);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.9 [33X[0;0YOpposite category[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQQ := HomalgFieldOfRationals();;[127X[104X
    [4X[25Xgap>[125X [27Xvec := MatrixCategory( QQ );;[127X[104X
    [4X[25Xgap>[125X [27Xop := Opposite( vec );;[127X[104X
    [4X[25Xgap>[125X [27XListKnownCategoricalProperties( op );[127X[104X
    [4X[28X[ "IsAbCategory", "IsAbelianCategory", "IsAbelianCategoryWithEnoughInjectives"[128X[104X
    [4X[28X    , "IsAbelianCategoryWithEnoughProjectives", "IsAdditiveCategory", [128X[104X
    [4X[28X  "IsBraidedMonoidalCategory", "IsClosedMonoidalCategory", [128X[104X
    [4X[28X  "IsCoclosedMonoidalCategory", "IsEnrichedOverCommutativeRegularSemigroup", [128X[104X
    [4X[28X  "IsEquippedWithHomomorphismStructure", "IsLinearCategoryOverCommutativeRing"[128X[104X
    [4X[28X    , [128X[104X
    [4X[28X  "IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms", [128X[104X
    [4X[28X  "IsMonoidalCategory", "IsPreAbelianCategory", [128X[104X
    [4X[28X  "IsRigidSymmetricClosedMonoidalCategory", [128X[104X
    [4X[28X  "IsRigidSymmetricCoclosedMonoidalCategory", "IsSkeletalCategory", [128X[104X
    [4X[28X  "IsStrictMonoidalCategory", "IsSymmetricClosedMonoidalCategory", [128X[104X
    [4X[28X  "IsSymmetricCoclosedMonoidalCategory", "IsSymmetricMonoidalCategory" ][128X[104X
    [4X[25Xgap>[125X [27XV1 := Opposite( TensorUnit( vec ) );;[127X[104X
    [4X[25Xgap>[125X [27XV2 := DirectSum( V1, V1 );;[127X[104X
    [4X[25Xgap>[125X [27XV3 := DirectSum( V1, V2 );;[127X[104X
    [4X[25Xgap>[125X [27XV4 := DirectSum( V1, V3 );;[127X[104X
    [4X[25Xgap>[125X [27XV5 := DirectSum( V1, V4 );;[127X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( MorphismBetweenDirectSums( op, [ ], [ ], [ V1 ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( MorphismBetweenDirectSums( op, [ V1 ], [ [ ] ], [ ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xalpha13 := InjectionOfCofactorOfDirectSum( [ V1, V2 ], 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha14 := InjectionOfCofactorOfDirectSum( [ V1, V2, V1 ], 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha15 := InjectionOfCofactorOfDirectSum( [ V2, V1, V2 ], 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha23 := InjectionOfCofactorOfDirectSum( [ V2, V1 ], 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha24 := InjectionOfCofactorOfDirectSum( [ V1, V2, V1 ], 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha25 := InjectionOfCofactorOfDirectSum( [ V2, V2, V1 ], 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xmat := [[127X[104X
    [4X[25X>[125X [27X    [ alpha13, alpha14, alpha15 ],[127X[104X
    [4X[25X>[125X [27X    [ alpha23, alpha24, alpha25 ][127X[104X
    [4X[25X>[125X [27X];;[127X[104X
    [4X[25Xgap>[125X [27Xmor := MorphismBetweenDirectSums( mat );;[127X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( mor );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( Opposite( mor ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOne( UniversalMorphismFromImage( mor, [ CoastrictionToImage( mor ), ImageEmbedding( mor ) ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.10 [33X[0;0YPreComposeList and PostComposeList[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfield := HomalgFieldOfRationals( );;[127X[104X
    [4X[25Xgap>[125X [27XA := VectorSpaceObject( 1, field );;[127X[104X
    [4X[25Xgap>[125X [27XB := VectorSpaceObject( 2, field );;[127X[104X
    [4X[25Xgap>[125X [27XC := VectorSpaceObject( 3, field );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha := VectorSpaceMorphism( A, HomalgMatrix( [ [ 1, 0, 0 ] ], 1, 3, field ), C );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta := VectorSpaceMorphism( C, HomalgMatrix( [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ], 3, 2, field ), B );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PreCompose( alpha, beta ), PostCompose( beta, alpha ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOne( PreComposeList( A, [ ], A ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PreComposeList( A, [ alpha ], C ), alpha );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PreComposeList( A, [ alpha, beta ], B ), PreCompose( alpha, beta ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOne( PostComposeList( A, [ ], A ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PostComposeList( A, [ alpha ], C ), alpha );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( PostComposeList( A, [ beta, alpha ], B ), PostCompose( beta, alpha ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X15.11 [33X[0;0YTerminal category[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "MonoidalCategories", ">= 2024.06-01", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XT := TerminalCategoryWithMultipleObjects( );[127X[104X
    [4X[28XTerminalCategoryWithMultipleObjects( )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( T );[127X[104X
    [4X[28XA CAP category with name TerminalCategoryWithMultipleObjects( ):[128X[104X
    [4X[28X[128X[104X
    [4X[28X82 primitive operations were used to derive 391 operations for this category \[128X[104X
    [4X[28Xwhich algorithmically[128X[104X
    [4X[28X* IsCategoryWithDecidableColifts[128X[104X
    [4X[28X* IsCategoryWithDecidableLifts[128X[104X
    [4X[28X* IsEquippedWithHomomorphismStructure[128X[104X
    [4X[28X* IsLinearCategoryOverCommutativeRing[128X[104X
    [4X[28X* IsLeftClosedMonoidalCategory[128X[104X
    [4X[28X* IsLeftCoclosedMonoidalCategory[128X[104X
    [4X[28X* IsAbelianCategoryWithEnoughInjectives[128X[104X
    [4X[28X* IsAbelianCategoryWithEnoughProjectives[128X[104X
    [4X[28X* IsRigidSymmetricClosedMonoidalCategory[128X[104X
    [4X[28X* IsRigidSymmetricCoclosedMonoidalCategory[128X[104X
    [4X[28Xand not yet algorithmically[128X[104X
    [4X[28X* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms[128X[104X
    [4X[28Xand furthermore mathematically[128X[104X
    [4X[28X* IsLocallyOfFiniteInjectiveDimension[128X[104X
    [4X[28X* IsLocallyOfFiniteProjectiveDimension[128X[104X
    [4X[28X* IsTerminalCategory[128X[104X
    [4X[25Xgap>[125X [27Xi := InitialObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27Xt := TerminalObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27Xz := ZeroObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( i );[127X[104X
    [4X[28XZeroObject[128X[104X
    [4X[25Xgap>[125X [27XDisplay( t );[127X[104X
    [4X[28XZeroObject[128X[104X
    [4X[25Xgap>[125X [27XDisplay( z );[127X[104X
    [4X[28XZeroObject[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( i, z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( t, z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xid_z := IdentityMorphism( z );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27Xfn_z := ZeroObjectFunctorial( T );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( id_z, fn_z );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( id_z, fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xa := "a" / T;[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( a );[127X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( a );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xaa := ObjectConstructor( T, "a" );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( aa );[127X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForObjects( a, aa );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicForObjects( a, aa );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism( SomeIsomorphismBetweenObjects( a, aa ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xb := "b" / T;[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( b );[127X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForObjects( a, b );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicForObjects( a, b );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xmor_ab := SomeIsomorphismBetweenObjects( a, b );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism( mor_ab );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplay( mor_ab );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| SomeIsomorphismBetweenObjects[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XHom_ab := MorphismsOfExternalHom( a, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( Hom_ab );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XHom_ab[1];[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( Hom_ab[1] );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XHom_ab[1] = mor_ab;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XHomStructure( mor_ab );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27Xt := TensorProduct( a, b );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( t );[127X[104X
    [4X[28XTensorProductOnObjects[128X[104X
    [4X[25Xgap>[125X [27Xa = t;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XTensorProduct( a, a ) = t;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xm := MorphismConstructor( a, "m", b );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| m[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( m );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn := MorphismConstructor( a, "n", b );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( n );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| n[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( m, n );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( m, n );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xm = n;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xhom_mn := HomStructure( m, n );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27Xid := IdentityMorphism( a );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( id );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| IdentityMorphism[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27Xm = id;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xid = MorphismConstructor( a, "xyz", a );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xzero := ZeroMorphism( a, a );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( zero );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| ZeroMorphism[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27Xid = zero;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLiftable( m, n );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xlift := Lift( m, n );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( lift );[127X[104X
    [4X[28Xa[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| Lift[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xa[128X[104X
    [4X[25Xgap>[125X [27XIsColiftable( m, n );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcolift := Colift( m, n );[127X[104X
    [4X[28X<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( colift );[127X[104X
    [4X[28Xb[128X[104X
    [4X[28X|[128X[104X
    [4X[28X| Colift[128X[104X
    [4X[28Xv[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XDirectProduct( T, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XEqualizer( T, z, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XCoproduct( T, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
    [4X[25Xgap>[125X [27XCoequalizer( T, z, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithMultipleObjects( )>[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "MonoidalCategories", ">= 2024.06-01", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XT := TerminalCategoryWithSingleObject( );[127X[104X
    [4X[28XTerminalCategoryWithSingleObject( )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( T );[127X[104X
    [4X[28XA CAP category with name TerminalCategoryWithSingleObject( ):[128X[104X
    [4X[28X[128X[104X
    [4X[28X76 primitive operations were used to derive 391 operations for this category \[128X[104X
    [4X[28Xwhich algorithmically[128X[104X
    [4X[28X* IsCategoryWithDecidableColifts[128X[104X
    [4X[28X* IsCategoryWithDecidableLifts[128X[104X
    [4X[28X* IsEquippedWithHomomorphismStructure[128X[104X
    [4X[28X* IsLinearCategoryOverCommutativeRing[128X[104X
    [4X[28X* IsLeftClosedMonoidalCategory[128X[104X
    [4X[28X* IsLeftCoclosedMonoidalCategory[128X[104X
    [4X[28X* IsAbelianCategoryWithEnoughInjectives[128X[104X
    [4X[28X* IsAbelianCategoryWithEnoughProjectives[128X[104X
    [4X[28X* IsRigidSymmetricClosedMonoidalCategory[128X[104X
    [4X[28X* IsRigidSymmetricCoclosedMonoidalCategory[128X[104X
    [4X[28Xand not yet algorithmically[128X[104X
    [4X[28X* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms[128X[104X
    [4X[28Xand furthermore mathematically[128X[104X
    [4X[28X* IsLocallyOfFiniteInjectiveDimension[128X[104X
    [4X[28X* IsLocallyOfFiniteProjectiveDimension[128X[104X
    [4X[28X* IsSkeletalCategory[128X[104X
    [4X[28X* IsStrictMonoidalCategory[128X[104X
    [4X[28X* IsTerminalCategory[128X[104X
    [4X[25Xgap>[125X [27Xi := InitialObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27Xt := TerminalObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27Xz := ZeroObject( T );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( i );[127X[104X
    [4X[28XA zero object in TerminalCategoryWithSingleObject( ).[128X[104X
    [4X[25Xgap>[125X [27XDisplay( t );[127X[104X
    [4X[28XA zero object in TerminalCategoryWithSingleObject( ).[128X[104X
    [4X[25Xgap>[125X [27XDisplay( z );[127X[104X
    [4X[28XA zero object in TerminalCategoryWithSingleObject( ).[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( i, z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( t, z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xid_z := IdentityMorphism( z );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27Xfn_z := ZeroObjectFunctorial( T );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XIsWellDefined( fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsEqualForMorphisms( id_z, fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( id_z, fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLiftable( id_z, fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLift( id_z, fn_z );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XIsColiftable( id_z, fn_z );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XColift( id_z, fn_z );[127X[104X
    [4X[28X<A zero, identity morphism in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XDirectProduct( T, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XEqualizer( T, z, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XCoproduct( T, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
    [4X[25Xgap>[125X [27XCoequalizer( T, z, [ ] );[127X[104X
    [4X[28X<A zero object in TerminalCategoryWithSingleObject( )>[128X[104X
  [4X[32X[104X
  
  
  [1X15.12 [33X[0;0YWrapperCategory[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "LinearAlgebraForCAP", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XQ := HomalgFieldOfRationals( );[127X[104X
    [4X[28XQ[128X[104X
    [4X[25Xgap>[125X [27XQmat := MATRIX_CATEGORY( Q );[127X[104X
    [4X[28XCategory of matrices over Q[128X[104X
    [4X[25Xgap>[125X [27XWrapper := WrapperCategory( Qmat, rec( ) );[127X[104X
    [4X[28XWrapperCategory( Category of matrices over Q )[128X[104X
    [4X[25Xgap>[125X [27Xmor := ZeroMorphism( ZeroObject( Wrapper ), ZeroObject( Wrapper ) );;[127X[104X
    [4X[25Xgap>[125X [27X2 * mor;;[127X[104X
    [4X[25Xgap>[125X [27XBasisOfExternalHom( Source( mor ), Range( mor ) );;[127X[104X
    [4X[25Xgap>[125X [27XCoefficientsOfMorphism( mor );;[127X[104X
    [4X[25Xgap>[125X [27Xdistinguished_object := DistinguishedObjectOfHomomorphismStructure( Wrapper );;[127X[104X
    [4X[25Xgap>[125X [27Xobject := HomomorphismStructureOnObjects( Source( mor ), Source( mor ) );;[127X[104X
    [4X[25Xgap>[125X [27XHomomorphismStructureOnMorphisms( mor, mor );;[127X[104X
    [4X[25Xgap>[125X [27XHomomorphismStructureOnMorphismsWithGivenObjects( object, mor, mor, object );;[127X[104X
    [4X[25Xgap>[125X [27Xiota := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( mor );;[127X[104X
    [4X[25Xgap>[125X [27XInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects( distinguished_object, mor, object );;[127X[104X
    [4X[25Xgap>[125X [27Xbeta := InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( mor ), Range( mor ), iota );;[127X[104X
    [4X[25Xgap>[125X [27XIsCongruentForMorphisms( mor, beta );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
