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15 Examples and Tests
 15.1 Dummy implementations

  15.1-1 Dummy categories
  15.1-2 Dummy rings
 15.2 Functors
 15.3 HandlePrecompiledTowers
 15.4 Homomorphism structure
 15.5 Homology object
 15.6 Liftable
 15.7 Monoidal Categories
 15.8 MorphismFromSourceToPushout and MorphismFromFiberProductToSink
 15.9 Opposite category
 15.10 PreComposeList and PostComposeList
 15.11 Terminal category
 15.12 WrapperCategory

15 Examples and Tests

15.1 Dummy implementations

15.1-1 Dummy categories
gap> LoadPackage( "CAP", false );
true
gap> list_of_operations_to_install := [
>     "ObjectConstructor",
>     "MorphismConstructor",
>     "ObjectDatum",
>     "MorphismDatum",
>     "IsCongruentForMorphisms",
>     "PreCompose",
>     "IdentityMorphism",
>     "DirectSum",
> ];;
gap> dummy := DummyCategory( rec(
>     list_of_operations_to_install := list_of_operations_to_install,
>     properties := [ "IsAdditiveCategory" ],
> ) );;
gap> ForAll( list_of_operations_to_install, o -> CanCompute( dummy, o ) );
true
gap> IsAdditiveCategory( dummy );
true

15.1-2 Dummy rings
gap> LoadPackage( "CAP", false );
true
gap> DummyRing( );
Dummy ring 1
gap> DummyRing( );
Dummy ring 2
gap> IsRing( DummyRing( ) );
true
gap> DummyCommutativeRing( );
Dummy commutative ring 1
gap> DummyCommutativeRing( );
Dummy commutative ring 2
gap> IsRing( DummyCommutativeRing( ) );
true
gap> IsCommutative( DummyCommutativeRing( ) );
true
gap> DummyField( );
Dummy field 1
gap> DummyField( );
Dummy field 2
gap> IsRing( DummyField( ) );
true
gap> IsField( DummyField( ) );
true

15.2 Functors

We create a binary functor F with one covariant and one contravariant component in two ways. Here is the first way to model a binary functor:

gap> field := HomalgFieldOfRationals( );;
gap> vec := LeftPresentations( field );;
gap> F := CapFunctor( "CohomForVec", [ vec, [ vec, true ] ], vec );;
gap> obj_func := function( A, B ) return TensorProductOnObjects( A, DualOnObjects( B ) ); end;;
gap> mor_func := function( source, alpha, beta, range ) return TensorProductOnMorphismsWithGivenTensorProducts( source, alpha, DualOnMorphisms( beta ), range ); end;;
gap> AddObjectFunction( F, obj_func );;
gap> AddMorphismFunction( F, mor_func );;

CAP regards F as a binary functor on a technical level, as we can see by looking at its input signature:

gap> InputSignature( F );
[ [ Category of left presentations of Q, false ], [ Category of left presentations of Q, true ] ]

We can see that ApplyFunctor works both on two arguments and on one argument (in the product category).

gap> V1 := TensorUnit( vec );;
gap> V3 := DirectSum( V1, V1, V1 );;
gap> pi1 := ProjectionInFactorOfDirectSum( [ V1, V1 ], 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( [ V3, V1 ], 1 );;
gap> value1 := ApplyFunctor( F, pi1, pi2 );;
gap> input := Product( pi1, Opposite( pi2 ) );;
gap> value2 := ApplyFunctor( F, input );;
gap> IsCongruentForMorphisms( value1, value2 );
true

Here is the second way to model a binary functor:

gap> F2 := CapFunctor( "CohomForVec2", Product( vec, Opposite( vec ) ), vec );;
gap> AddObjectFunction( F2, a -> obj_func( a[1], Opposite( a[2] ) ) );;
gap> AddMorphismFunction( F2, function( source, datum, range ) return mor_func( source, datum[1], Opposite( datum[2] ), range ); end );;
gap> value3 := ApplyFunctor( F2,input );;
gap> IsCongruentForMorphisms( value1, value3 );
true

CAP regards F2 as a unary functor on a technical level, as we can see by looking at its input signature:

gap> InputSignature( F2 );
[ [ Product of: Category of left presentations of Q, Opposite( Category of left presentations of Q ), false ] ]

Installation of the first functor as a GAP-operation. It will be installed both as a unary and binary version.

gap> InstallFunctor( F, "F_installation" );;
gap> F_installation( pi1, pi2 );;
gap> F_installation( input );;
gap> F_installationOnObjects( V1, V1 );;
gap> F_installationOnObjects( Product( V1, Opposite( V1 ) ) );;
gap> F_installationOnMorphisms( pi1, pi2 );;
gap> F_installationOnMorphisms( input );;

Installation of the second functor as a GAP-operation. It will be installed only as a unary version.

gap> InstallFunctor( F2, "F_installation2" );;
gap> F_installation2( input );;
gap> F_installation2OnObjects( Product( V1, Opposite( V1 ) ) );;
gap> F_installation2OnMorphisms( input );;

15.3 HandlePrecompiledTowers

gap> LoadPackage( "CAP", false );
true
gap> dummy1 := CreateCapCategory( );;
gap> dummy2 := CreateCapCategory( );;
gap> dummy3 := CreateCapCategory( );;
gap> PrintAndReturn := function ( string )
>     Print( string, "\n" ); return string; end;;
gap> dummy1!.compiler_hints := rec( );;
gap> dummy1!.compiler_hints.precompiled_towers := [
>   rec(
>     remaining_constructors_in_tower := [ "Constructor1" ],
>     precompiled_functions_adder := cat ->
>       PrintAndReturn( "Adding precompiled operations for Constructor1" ),
>   ),
>   rec(
>     remaining_constructors_in_tower := [ "Constructor1", "Constructor2" ],
>     precompiled_functions_adder := cat ->
>       PrintAndReturn( "Adding precompiled operations for Constructor2" ),
>   ),
> ];;
gap> HandlePrecompiledTowers( dummy2, dummy1, "Constructor1" );
Adding precompiled operations for Constructor1
gap> HandlePrecompiledTowers( dummy3, dummy2, "Constructor2" );
Adding precompiled operations for Constructor2

15.4 Homomorphism structure

gap> ReadPackage( "CAP", "examples/FieldAsCategory.g" );;
gap> Q := HomalgFieldOfRationals();;
gap> Qoid := FieldAsCategory( Q );;
gap> a := FieldAsCategoryMorphism( Qoid, 1/2 );;
gap> b := FieldAsCategoryMorphism( Qoid, -2/3 );;
gap> u := FieldAsCategoryUniqueObject( Qoid );;
gap> IsCongruentForMorphisms( a,
>     InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism(
>         u,u,
>         InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure(
>             a
>         )
>     )
> );
true
gap> a = HomStructure( u, u, HomStructure( a ) );
true
gap> IsEqualForObjects( HomStructure( Qoid ), DistinguishedObjectOfHomomorphismStructure( Qoid ) );
true
gap> c := FieldAsCategoryMorphism( Qoid, 3 );;
gap> d := FieldAsCategoryMorphism( Qoid, 0 );;
gap> left_coeffs := [ [ a, b ], [ c, d ] ];;
gap> right_coeffs := [ [ PreCompose( a, b ), PreCompose( b, c ) ], [ c, PreCompose( a, a ) ] ];;
gap> right_side := [ a, b ];;
gap> MereExistenceOfSolutionOfLinearSystemInAbCategory( left_coeffs, right_coeffs, right_side );
true
gap> solution := 
>     SolveLinearSystemInAbCategory(
>     left_coeffs,
>     right_coeffs,
>     right_side
> );;
gap> ForAll( [ 1, 2 ], i ->
>     IsCongruentForMorphisms(
>         Sum( List( [ 1, 2 ], j -> PreCompose( [ left_coeffs[i][j], solution[j], right_coeffs[i][j] ] ) ) ),
>         right_side[i]
>     )
> );
true
gap> IsLiftable( c, d );
false
gap> LiftOrFail( c, d );
fail
gap> IsLiftable( d, c );
true
gap> LiftOrFail( d, c );
0
gap> Lift( d, c );
0
gap> IsColiftable( c, d );
true
gap> ColiftOrFail( c, d );
0
gap> Colift( c, d );
0
gap> IsColiftable( d, c );
false
gap> ColiftOrFail( d, c );
fail

gap> ReadPackage( "CAP", "examples/StringsAsCategory.g" );;
gap> C := StringsAsCategory();;
gap> obj1 := StringsAsCategoryObject( C, "qaeiou" );;
gap> obj2 := StringsAsCategoryObject( C, "qxayeziouT" );;
gap> mor := StringsAsCategoryMorphism( C, obj1, "xyzaTe", obj2 );;
gap> IsWellDefined( mor );
true
gap> ## Test SimplifyObject
> IsEqualForObjects( SimplifyObject( obj1, 0 ), obj1 );
true
gap> IsEqualForObjects( SimplifyObject( obj1, 1 ), obj1 );
false
gap> ForAny( [0,1,2,3,4], i -> IsEqualForObjects( SimplifyObject( obj1, i ), SimplifyObject( obj1, i + 1 ) ) );
false
gap> ForAll( [5,6,7,8], i -> IsEqualForObjects( SimplifyObject( obj1, i ), SimplifyObject( obj1, i + 1 ) ) );
true
gap> ## Test SimplifyMorphism
> IsEqualForMorphisms( SimplifyMorphism( mor, 0 ), mor );
true
gap> IsEqualForMorphisms( SimplifyMorphism( mor, 1 ), mor );
false
gap> ForAny( [0,1], i -> IsEqualForMorphisms( SimplifyMorphism( mor, i ), SimplifyMorphism( mor, i + 1 ) ) );
false
gap> ForAll( [2,3,4,5], i -> IsEqualForMorphisms( SimplifyMorphism( mor, i ), SimplifyMorphism( mor, i + 1 ) ) );
true
gap> ## Test SimplifySource
> IsEqualForMorphismsOnMor( SimplifySource( mor, 0 ), mor );
true
gap> IsEqualForMorphismsOnMor( SimplifySource( mor, 1 ), mor );
false
gap> ForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySource( mor, i ), SimplifySource( mor, i + 1 ) ) );
false
gap> ForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySource( mor, i ), SimplifySource( mor, i + 1 ) ) );
true
gap> IsCongruentForMorphisms(
>     PreCompose( SimplifySource_IsoFromInputObject( mor, infinity ), SimplifySource( mor, infinity ) ), mor
> );
true
gap> IsCongruentForMorphisms(
>     PreCompose( SimplifySource_IsoToInputObject( mor, infinity ), mor ) , SimplifySource( mor, infinity )
> );
true
gap> ## Test SimplifyRange
> IsEqualForMorphismsOnMor( SimplifyRange( mor, 0 ), mor );
true
gap> IsEqualForMorphismsOnMor( SimplifyRange( mor, 1 ), mor );
false
gap> ForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifyRange( mor, i ), SimplifyRange( mor, i + 1 ) ) );
false
gap> ForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifyRange( mor, i ), SimplifyRange( mor, i + 1 ) ) );
true
gap> IsCongruentForMorphisms(
>     PreCompose( SimplifyRange( mor, infinity ), SimplifyRange_IsoToInputObject( mor, infinity ) ), mor
> );
true
gap> IsCongruentForMorphisms(
>     PreCompose( mor, SimplifyRange_IsoFromInputObject( mor, infinity ) ), SimplifyRange( mor, infinity )
> );
true
gap> ## Test SimplifySourceAndRange
> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, 0 ), mor );
true
gap> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, 1 ), mor );
false
gap> ForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, i ), SimplifySourceAndRange( mor, i + 1 ) ) );
false
gap> ForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( mor, i ), SimplifySourceAndRange( mor, i + 1 ) ) );
true
gap> IsCongruentForMorphisms(
>     mor,
>     PreCompose( [ SimplifySourceAndRange_IsoFromInputSource( mor, infinity ),
>                   SimplifySourceAndRange( mor, infinity ),
>                   SimplifySourceAndRange_IsoToInputRange( mor, infinity ) ] )
> );
true
gap> IsCongruentForMorphisms(
>     SimplifySourceAndRange( mor, infinity ),
>     PreCompose( [ SimplifySourceAndRange_IsoToInputSource( mor, infinity ),
>                   mor,
>                   SimplifySourceAndRange_IsoFromInputRange( mor, infinity ) ] )
> );
true
gap> ## Test SimplifyEndo
> endo := StringsAsCategoryMorphism( C, obj1, "uoiea", obj1 );;
gap> IsWellDefined( endo );
true
gap> IsEqualForMorphismsOnMor( SimplifyEndo( endo, 0 ), endo );
true
gap> IsEqualForMorphismsOnMor( SimplifyEndo( endo, 1 ), endo );
false
gap> ForAny( [0,1,2,3,4], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( endo, i ), SimplifySourceAndRange( endo, i + 1 ) ) );
false
gap> ForAll( [5,6,7,8,9], i -> IsEqualForMorphismsOnMor( SimplifySourceAndRange( endo, i ), SimplifySourceAndRange( endo, i + 1 ) ) );
true
gap> iota := SimplifyEndo_IsoToInputObject( endo, infinity );;
gap> iota_inv := SimplifyEndo_IsoFromInputObject( endo, infinity );;
gap> IsCongruentForMorphisms( PreCompose( [ iota_inv, SimplifyEndo( endo, infinity ), iota ] ), endo );
true

15.5 Homology object

gap> field := HomalgFieldOfRationals( );;
gap> A := VectorSpaceObject( 1, field );;
gap> B := VectorSpaceObject( 2, field );;
gap> C := VectorSpaceObject( 3, field );;
gap> alpha := VectorSpaceMorphism( A, HomalgMatrix( [ [ 1, 0, 0 ] ], 1, 3, field ), C );;
gap> beta := VectorSpaceMorphism( C, HomalgMatrix( [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ], 3, 2, field ), B );;
gap> IsZero( PreCompose( alpha, beta ) );
false
gap> IsCongruentForMorphisms(
>     IdentityMorphism( HomologyObject( alpha, beta ) ),
>     HomologyObjectFunctorial( alpha, beta, IdentityMorphism( C ), alpha, beta )
> );
true
gap> kernel_beta := KernelEmbedding( beta );;
gap> K := Source( kernel_beta );;
gap> IsIsomorphism(
>     HomologyObjectFunctorial( 
>         MorphismFromZeroObject( K ), 
>         MorphismIntoZeroObject( K ),
>         kernel_beta,
>         MorphismFromZeroObject( Source( beta ) ),
>         beta
>     )
> );
true
gap> cokernel_alpha := CokernelProjection( alpha );;
gap> Co := Range( cokernel_alpha );;
gap> IsIsomorphism(
>     HomologyObjectFunctorial( 
>         alpha,
>         MorphismIntoZeroObject( Range( alpha ) ),
>         cokernel_alpha,
>         MorphismFromZeroObject( Co ),
>         MorphismIntoZeroObject( Co )
>     )
> );
true
gap> alpha_op := Opposite( alpha );;
gap> beta_op := Opposite( beta );;
gap> IsCongruentForMorphisms(
>     IdentityMorphism( HomologyObject( beta_op, alpha_op ) ),
>     HomologyObjectFunctorial( beta_op, alpha_op, IdentityMorphism( Opposite( C ) ), beta_op, alpha_op )
> );
true
gap> kernel_beta := KernelEmbedding( beta_op );;
gap> K := Source( kernel_beta );;
gap> IsIsomorphism(
>     HomologyObjectFunctorial( 
>         MorphismFromZeroObject( K ), 
>         MorphismIntoZeroObject( K ),
>         kernel_beta,
>         MorphismFromZeroObject( Source( beta_op ) ),
>         beta_op
>     )
> );
true
gap> cokernel_alpha := CokernelProjection( alpha_op );;
gap> Co := Range( cokernel_alpha );;
gap> IsIsomorphism(
>     HomologyObjectFunctorial( 
>         alpha_op,
>         MorphismIntoZeroObject( Range( alpha_op ) ),
>         cokernel_alpha,
>         MorphismFromZeroObject( Co ),
>         MorphismIntoZeroObject( Co )
>     )
> );
true

15.6 Liftable

gap> field := HomalgFieldOfRationals( );;
gap> V := VectorSpaceObject( 1, field );;
gap> W := VectorSpaceObject( 2, field );;
gap> alpha := VectorSpaceMorphism( V, HomalgMatrix( [ [ 1, -1 ] ], 1, 2, field ), W );;
gap> beta := VectorSpaceMorphism( W, HomalgMatrix( [ [ 1, 2 ], [ 3, 4 ] ], 2, 2, field ), W );;
gap> IsLiftable( alpha, beta );
true
gap> IsLiftable( beta, alpha );
false
gap> IsLiftableAlongMonomorphism( beta, alpha );
true
gap> gamma := VectorSpaceMorphism( W, HomalgMatrix( [ [ 1 ], [ 1 ] ], 2, 1, field ), V );;
gap> IsColiftable( beta, gamma );
true
gap> IsColiftable( gamma, beta );
false
gap> IsColiftableAlongEpimorphism( beta, gamma );
true
gap> PreCompose( PreInverseForMorphisms( gamma ), gamma ) = IdentityMorphism( V );
true
gap> PreCompose( alpha, PostInverseForMorphisms( alpha ) ) = IdentityMorphism( V );
true

15.7 Monoidal Categories

gap> ZZZ := HomalgRingOfIntegers();;
gap> Ml := AsLeftPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, ZZZ ) );
<An object in Category of left presentations of Z>
gap> Nl := AsLeftPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZZ ) );
<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 ) );
[ [  1,  2 ],
  [  0,  3 ] ]
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 ) );
[ [  -3 ],
  [   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 ],
  [  -1 ] ]
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, ZZZ ) );
<An object in Category of right presentations of Z>
gap> Nr := AsRightPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, ZZZ ) );
<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 ) );
[ [  1,  0 ],
  [  2,  3 ] ]
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 ) );
[ [  -3,   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 ],
  [  -1 ] ]
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

15.8 MorphismFromSourceToPushout and MorphismFromFiberProductToSink

gap> field := HomalgFieldOfRationals( );;
gap> A := VectorSpaceObject( 3, field );;
gap> B := VectorSpaceObject( 2, field );;
gap> alpha := VectorSpaceMorphism( B, HomalgMatrix( [ [ 1, -1, 1 ], [ 1, 1, 1 ] ], 2, 3, field ), A );;
gap> beta := VectorSpaceMorphism( B, HomalgMatrix( [ [ 1, 2, 1 ], [ 2, 1, 1 ] ], 2, 3, field ), A );;
gap> m := MorphismFromFiberProductToSink( [ alpha, beta ] );;
gap> IsCongruentForMorphisms(
>     m,
>     PreCompose( ProjectionInFactorOfFiberProduct( [ alpha, beta ], 1 ), alpha )
> );
true
gap> IsCongruentForMorphisms(
>     m,
>     PreCompose( ProjectionInFactorOfFiberProduct( [ alpha, beta ], 2 ), beta )
> );
true
gap> IsCongruentForMorphisms(
> MorphismFromKernelObjectToSink( alpha ),
>     PreCompose( KernelEmbedding( alpha ), alpha )
> );
true
gap> alpha_p := DualOnMorphisms( alpha );;
gap> beta_p := DualOnMorphisms( beta );;
gap> m_p := MorphismFromSourceToPushout( [ alpha_p, beta_p ] );;
gap> IsCongruentForMorphisms(
>     m_p,
>     PreCompose( alpha_p, InjectionOfCofactorOfPushout( [ alpha_p, beta_p ], 1 ) )
> );
true
gap> IsCongruentForMorphisms(
>     m_p,
>     PreCompose( beta_p, InjectionOfCofactorOfPushout( [ alpha_p, beta_p ], 2 ) )
> );
true
gap> IsCongruentForMorphisms(
>     MorphismFromSourceToCokernelObject( alpha_p ),
>     PreCompose( alpha_p, CokernelProjection( alpha_p ) )
> );
true

15.9 Opposite category

gap> QQ := HomalgFieldOfRationals();;
gap> vec := MatrixCategory( QQ );;
gap> op := Opposite( vec );;
gap> ListKnownCategoricalProperties( op );
[ "IsAbCategory", "IsAbelianCategory", "IsAbelianCategoryWithEnoughInjectives"
    , "IsAbelianCategoryWithEnoughProjectives", "IsAdditiveCategory", 
  "IsBraidedMonoidalCategory", "IsClosedMonoidalCategory", 
  "IsCoclosedMonoidalCategory", "IsEnrichedOverCommutativeRegularSemigroup", 
  "IsEquippedWithHomomorphismStructure", "IsLinearCategoryOverCommutativeRing"
    , 
  "IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms", 
  "IsMonoidalCategory", "IsPreAbelianCategory", 
  "IsRigidSymmetricClosedMonoidalCategory", 
  "IsRigidSymmetricCoclosedMonoidalCategory", "IsSkeletalCategory", 
  "IsStrictMonoidalCategory", "IsSymmetricClosedMonoidalCategory", 
  "IsSymmetricCoclosedMonoidalCategory", "IsSymmetricMonoidalCategory" ]
gap> V1 := Opposite( TensorUnit( vec ) );;
gap> V2 := DirectSum( V1, V1 );;
gap> V3 := DirectSum( V1, V2 );;
gap> V4 := DirectSum( V1, V3 );;
gap> V5 := DirectSum( V1, V4 );;
gap> IsWellDefined( MorphismBetweenDirectSums( op, [ ], [ ], [ V1 ] ) );
true
gap> IsWellDefined( MorphismBetweenDirectSums( op, [ V1 ], [ [ ] ], [ ] ) );
true
gap> alpha13 := InjectionOfCofactorOfDirectSum( [ V1, V2 ], 1 );;
gap> alpha14 := InjectionOfCofactorOfDirectSum( [ V1, V2, V1 ], 3 );;
gap> alpha15 := InjectionOfCofactorOfDirectSum( [ V2, V1, V2 ], 2 );;
gap> alpha23 := InjectionOfCofactorOfDirectSum( [ V2, V1 ], 1 );;
gap> alpha24 := InjectionOfCofactorOfDirectSum( [ V1, V2, V1 ], 2 );;
gap> alpha25 := InjectionOfCofactorOfDirectSum( [ V2, V2, V1 ], 1 );;
gap> mat := [
>     [ alpha13, alpha14, alpha15 ],
>     [ alpha23, alpha24, alpha25 ]
> ];;
gap> mor := MorphismBetweenDirectSums( mat );;
gap> IsWellDefined( mor );
true
gap> IsWellDefined( Opposite( mor ) );
true
gap> IsOne( UniversalMorphismFromImage( mor, [ CoastrictionToImage( mor ), ImageEmbedding( mor ) ] ) );
true

15.10 PreComposeList and PostComposeList

gap> field := HomalgFieldOfRationals( );;
gap> A := VectorSpaceObject( 1, field );;
gap> B := VectorSpaceObject( 2, field );;
gap> C := VectorSpaceObject( 3, field );;
gap> alpha := VectorSpaceMorphism( A, HomalgMatrix( [ [ 1, 0, 0 ] ], 1, 3, field ), C );;
gap> beta := VectorSpaceMorphism( C, HomalgMatrix( [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ], 3, 2, field ), B );;
gap> IsCongruentForMorphisms( PreCompose( alpha, beta ), PostCompose( beta, alpha ) );
true
gap> IsOne( PreComposeList( A, [ ], A ) );
true
gap> IsCongruentForMorphisms( PreComposeList( A, [ alpha ], C ), alpha );
true
gap> IsCongruentForMorphisms( PreComposeList( A, [ alpha, beta ], B ), PreCompose( alpha, beta ) );
true
gap> IsOne( PostComposeList( A, [ ], A ) );
true
gap> IsCongruentForMorphisms( PostComposeList( A, [ alpha ], C ), alpha );
true
gap> IsCongruentForMorphisms( PostComposeList( A, [ beta, alpha ], B ), PostCompose( beta, alpha ) );
true

15.11 Terminal category

gap> LoadPackage( "MonoidalCategories", ">= 2024.06-01", false );
true
gap> T := TerminalCategoryWithMultipleObjects( );
TerminalCategoryWithMultipleObjects( )
gap> Display( T );
A CAP category with name TerminalCategoryWithMultipleObjects( ):

82 primitive operations were used to derive 391 operations for this category \
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRing
* IsLeftClosedMonoidalCategory
* IsLeftCoclosedMonoidalCategory
* IsAbelianCategoryWithEnoughInjectives
* IsAbelianCategoryWithEnoughProjectives
* IsRigidSymmetricClosedMonoidalCategory
* IsRigidSymmetricCoclosedMonoidalCategory
and not yet algorithmically
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
and furthermore mathematically
* IsLocallyOfFiniteInjectiveDimension
* IsLocallyOfFiniteProjectiveDimension
* IsTerminalCategory
gap> i := InitialObject( T );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> t := TerminalObject( T );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> z := ZeroObject( T );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Display( i );
ZeroObject
gap> Display( t );
ZeroObject
gap> Display( z );
ZeroObject
gap> IsIdenticalObj( i, z );
true
gap> IsIdenticalObj( t, z );
true
gap> id_z := IdentityMorphism( z );
<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>
gap> fn_z := ZeroObjectFunctorial( T );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> IsEqualForMorphisms( id_z, fn_z );
false
gap> IsCongruentForMorphisms( id_z, fn_z );
true
gap> a := "a" / T;
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Display( a );
a
gap> IsWellDefined( a );
true
gap> aa := ObjectConstructor( T, "a" );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Display( aa );
a
gap> IsEqualForObjects( a, aa );
true
gap> IsIsomorphicForObjects( a, aa );
true
gap> IsIsomorphism( SomeIsomorphismBetweenObjects( a, aa ) );
true
gap> b := "b" / T;
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Display( b );
b
gap> IsEqualForObjects( a, b );
false
gap> IsIsomorphicForObjects( a, b );
true
gap> mor_ab := SomeIsomorphismBetweenObjects( a, b );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> IsIsomorphism( mor_ab );
true
gap> Display( mor_ab );
a
|
| SomeIsomorphismBetweenObjects
v
b
gap> Hom_ab := MorphismsOfExternalHom( a, b );;
gap> Length( Hom_ab );
1
gap> Hom_ab[1];
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( Hom_ab[1] );
a
|
| InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
v
b
gap> Hom_ab[1] = mor_ab;
true
gap> HomStructure( mor_ab );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> t := TensorProduct( a, b );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Display( t );
TensorProductOnObjects
gap> a = t;
false
gap> TensorProduct( a, a ) = t;
true
gap> m := MorphismConstructor( a, "m", b );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( m );
a
|
| m
v
b
gap> IsWellDefined( m );
true
gap> n := MorphismConstructor( a, "n", b );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( n );
a
|
| n
v
b
gap> IsEqualForMorphisms( m, n );
false
gap> IsCongruentForMorphisms( m, n );
true
gap> m = n;
true
gap> hom_mn := HomStructure( m, n );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> id := IdentityMorphism( a );
<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( id );
a
|
| IdentityMorphism
v
a
gap> m = id;
false
gap> id = MorphismConstructor( a, "xyz", a );
true
gap> zero := ZeroMorphism( a, a );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( zero );
a
|
| ZeroMorphism
v
a
gap> id = zero;
true
gap> IsLiftable( m, n );
true
gap> lift := Lift( m, n );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( lift );
a
|
| Lift
v
a
gap> IsColiftable( m, n );
true
gap> colift := Colift( m, n );
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
gap> Display( colift );
b
|
| Colift
v
b
gap> DirectProduct( T, [ ] );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Equalizer( T, z, [ ] );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Coproduct( T, [ ] );
<A zero object in TerminalCategoryWithMultipleObjects( )>
gap> Coequalizer( T, z, [ ] );
<A zero object in TerminalCategoryWithMultipleObjects( )>

gap> LoadPackage( "MonoidalCategories", ">= 2024.06-01", false );
true
gap> T := TerminalCategoryWithSingleObject( );
TerminalCategoryWithSingleObject( )
gap> Display( T );
A CAP category with name TerminalCategoryWithSingleObject( ):

76 primitive operations were used to derive 391 operations for this category \
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRing
* IsLeftClosedMonoidalCategory
* IsLeftCoclosedMonoidalCategory
* IsAbelianCategoryWithEnoughInjectives
* IsAbelianCategoryWithEnoughProjectives
* IsRigidSymmetricClosedMonoidalCategory
* IsRigidSymmetricCoclosedMonoidalCategory
and not yet algorithmically
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
and furthermore mathematically
* IsLocallyOfFiniteInjectiveDimension
* IsLocallyOfFiniteProjectiveDimension
* IsSkeletalCategory
* IsStrictMonoidalCategory
* IsTerminalCategory
gap> i := InitialObject( T );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> t := TerminalObject( T );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> z := ZeroObject( T );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> Display( i );
A zero object in TerminalCategoryWithSingleObject( ).
gap> Display( t );
A zero object in TerminalCategoryWithSingleObject( ).
gap> Display( z );
A zero object in TerminalCategoryWithSingleObject( ).
gap> IsIdenticalObj( i, z );
true
gap> IsIdenticalObj( t, z );
true
gap> IsWellDefined( z );
true
gap> id_z := IdentityMorphism( z );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> fn_z := ZeroObjectFunctorial( T );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> IsWellDefined( fn_z );
true
gap> IsEqualForMorphisms( id_z, fn_z );
true
gap> IsCongruentForMorphisms( id_z, fn_z );
true
gap> IsLiftable( id_z, fn_z );
true
gap> Lift( id_z, fn_z );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> IsColiftable( id_z, fn_z );
true
gap> Colift( id_z, fn_z );
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
gap> DirectProduct( T, [ ] );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> Equalizer( T, z, [ ] );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> Coproduct( T, [ ] );
<A zero object in TerminalCategoryWithSingleObject( )>
gap> Coequalizer( T, z, [ ] );
<A zero object in TerminalCategoryWithSingleObject( )>

15.12 WrapperCategory

gap> LoadPackage( "LinearAlgebraForCAP", false );
true
gap> Q := HomalgFieldOfRationals( );
Q
gap> Qmat := MATRIX_CATEGORY( Q );
Category of matrices over Q
gap> Wrapper := WrapperCategory( Qmat, rec( ) );
WrapperCategory( Category of matrices over Q )
gap> mor := ZeroMorphism( ZeroObject( Wrapper ), ZeroObject( Wrapper ) );;
gap> 2 * mor;;
gap> BasisOfExternalHom( Source( mor ), Range( mor ) );;
gap> CoefficientsOfMorphism( mor );;
gap> distinguished_object := DistinguishedObjectOfHomomorphismStructure( Wrapper );;
gap> object := HomomorphismStructureOnObjects( Source( mor ), Source( mor ) );;
gap> HomomorphismStructureOnMorphisms( mor, mor );;
gap> HomomorphismStructureOnMorphismsWithGivenObjects( object, mor, mor, object );;
gap> iota := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( mor );;
gap> InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects( distinguished_object, mor, object );;
gap> beta := InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( mor ), Range( mor ), iota );;
gap> IsCongruentForMorphisms( mor, beta );
true
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