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
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
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> ring := HomalgRingOfIntegers( );; gap> vec := LeftPresentations( ring );; 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 Z, false ], [ Category of left presentations of Z, 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 Z, Opposite( Category of left presentations of Z ), 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 );;
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
gap> LoadPackage( "CAP", false ); true gap> T := TerminalCategoryWithMultipleObjects( ); TerminalCategoryWithMultipleObjects( ) 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 ); InitialObject gap> Display( t ); TerminalObject gap> Display( z ); ZeroObject gap> IsIdenticalObj( i, z ); false gap> IsIdenticalObj( t, z ); false 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> 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( "CAP", false ); true gap> T := TerminalCategoryWithSingleObject( ); TerminalCategoryWithSingleObject( ) 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 ); false gap> IsIdenticalObj( t, z ); false 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( )>
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