Examples and Tests
Dummy implementations 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
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
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 );;
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
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
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
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
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
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
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
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
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( )>
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