Examples and Tests
Basic Commands
gap> Q := HomalgFieldOfRationals();; gap> a := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> HasIsProjective( a ) and IsProjective( a ); true gap> vec := MatrixCategory( Q );; gap> ap := 3/vec;; gap> IsEqualForObjects( a, ap ); true gap> b := VectorSpaceObject( 4, Q ); <A vector space object over Q of dimension 4> gap> homalg_matrix := HomalgMatrix( [ [ 1, 0, 0, 0 ], > [ 0, 1, 0, -1 ], > [ -1, 0, 2, 1 ] ], 3, 4, Q ); <A 3 x 4 matrix over an internal ring> gap> alpha := VectorSpaceMorphism( a, homalg_matrix, b ); <A morphism in Category of matrices over Q> gap> Display( alpha ); [ [ 1, 0, 0, 0 ], [ 0, 1, 0, -1 ], [ -1, 0, 2, 1 ] ] A morphism in Category of matrices over Q gap> alphap := homalg_matrix/vec;; gap> IsCongruentForMorphisms( alpha, alphap ); true gap> homalg_matrix := HomalgMatrix( [ [ 1, 1, 0, 0 ], > [ 0, 1, 0, -1 ], > [ -1, 0, 2, 1 ] ], 3, 4, Q ); <A 3 x 4 matrix over an internal ring> gap> beta := VectorSpaceMorphism( a, homalg_matrix, b ); <A morphism in Category of matrices over Q> gap> CokernelObject( alpha ); <A vector space object over Q of dimension 1> gap> c := CokernelProjection( alpha );; #@if ValueOption( "no_precompiled_code" ) <> true gap> Display( c ); [ [ 0 ], [ 1 ], [ -1/2 ], [ 1 ] ] A split epimorphism in Category of matrices over Q #@fi gap> gamma := UniversalMorphismIntoDirectSum( [ c, c ] );; gap> Display( gamma ); [ [ 0, 0 ], [ 1, 1 ], [ -1/2, -1/2 ], [ 1, 1 ] ] A morphism in Category of matrices over Q gap> colift := CokernelColift( alpha, gamma );; gap> IsEqualForMorphisms( PreCompose( c, colift ), gamma ); true gap> FiberProduct( alpha, beta ); <A vector space object over Q of dimension 2> gap> F := FiberProduct( alpha, beta ); <A vector space object over Q of dimension 2> gap> p1 := ProjectionInFactorOfFiberProduct( [ alpha, beta ], 1 ); <A morphism in Category of matrices over Q> gap> Display( PreCompose( p1, alpha ) ); [ [ 0, 1, 0, -1 ], [ -1, 0, 2, 1 ] ] A morphism in Category of matrices over Q gap> Pushout( alpha, beta ); <A vector space object over Q of dimension 5> gap> i1 := InjectionOfCofactorOfPushout( [ alpha, beta ], 1 ); <A morphism in Category of matrices over Q> gap> i2 := InjectionOfCofactorOfPushout( [ alpha, beta ], 2 ); <A morphism in Category of matrices over Q> gap> u := UniversalMorphismFromDirectSum( [ b, b ], [ i1, i2 ] ); <A morphism in Category of matrices over Q> gap> Display( u ); [ [ 0, 1, 1, 0, 0 ], [ 1, 0, 1, 0, -1 ], [ -1/2, 0, 1/2, 1, 1/2 ], [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ] A morphism in Category of matrices over Q gap> KernelObjectFunctorial( u, IdentityMorphism( Source( u ) ), u ) = IdentityMorphism( VectorSpaceObject( 3, Q ) ); true gap> IsZero( CokernelObjectFunctorial( u, IdentityMorphism( Range( u ) ), u ) ); true gap> DirectProductFunctorial( [ u, u ] ) = DirectSumFunctorial( [ u, u ] ); true gap> CoproductFunctorial( [ u, u ] ) = DirectSumFunctorial( [ u, u ] ); true gap> IsOne( FiberProductFunctorial( [ u, u ], [ IdentityMorphism( Source( u ) ), IdentityMorphism( Source( u ) ) ], [ u, u ] ) ); true gap> IsOne( PushoutFunctorial( [ u, u ], [ IdentityMorphism( Range( u ) ), IdentityMorphism( Range( u ) ) ], [ u, u ] ) ); true gap> IsCongruentForMorphisms( (1/2) * alpha, alpha * (1/2) ); true gap> Dimension( HomomorphismStructureOnObjects( a, b ) ) = Dimension( a ) * Dimension( b ); true gap> IsCongruentForMorphisms( > PreCompose( [ u, DualOnMorphisms( i1 ), DualOnMorphisms( alpha ) ] ), > InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( u ), Source( alpha ), > PreCompose( > InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( DualOnMorphisms( i1 ) ), > HomomorphismStructureOnMorphisms( u, DualOnMorphisms( alpha ) ) > ) > ) > ); true gap> alpha_op := Opposite( alpha ); <A morphism in Opposite( Category of matrices over Q )> gap> basis := BasisOfExternalHom( Source( alpha_op ), Range( alpha_op ) );; gap> coeffs := CoefficientsOfMorphism( alpha_op ); [ 1, 0, 0, 0, 0, 1, 0, -1, -1, 0, 2, 1 ] gap> IsEqualForMorphisms( alpha_op, coeffs * basis ); true gap> vec := CapCategory( alpha );; gap> t := TensorUnit( vec );; gap> z := ZeroObject( vec );; gap> IsCongruentForMorphisms( > ZeroObjectFunctorial( vec ), > InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( z, z, ZeroMorphism( t, z ) ) > ); true gap> IsCongruentForMorphisms( > ZeroObjectFunctorial( vec ), > InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( > z, z, > InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( ZeroObjectFunctorial( vec ) ) > ) > ); true gap> right_side := PreCompose( [ i1, DualOnMorphisms( u ), u ] );; gap> x := SolveLinearSystemInAbCategory( [ [ i1 ] ], [ [ u ] ], [ right_side ] )[1];; gap> IsCongruentForMorphisms( PreCompose( [ i1, x, u ] ), right_side ); true gap> a_otimes_b := TensorProductOnObjects( a, b ); <A vector space object over Q of dimension 12> gap> hom_ab := InternalHomOnObjects( a, b ); <A vector space object over Q of dimension 12> gap> cohom_ab := InternalCoHomOnObjects( a, b ); <A vector space object over Q of dimension 12> gap> hom_ab = cohom_ab; true gap> 1_ab := VectorSpaceMorphism( > a_otimes_b, > HomalgIdentityMatrix( Dimension( a_otimes_b ), Q ), > a_otimes_b > ); <A morphism in Category of matrices over Q> gap> 1_hom_ab := VectorSpaceMorphism( > hom_ab, > HomalgIdentityMatrix( Dimension( hom_ab ), Q ), > hom_ab > ); <A morphism in Category of matrices over Q> gap> 1_cohom_ab := VectorSpaceMorphism( > cohom_ab, > HomalgIdentityMatrix( Dimension( cohom_ab ), Q ), > cohom_ab > ); <A morphism in Category of matrices over Q> gap> ev_ab := ClosedMonoidalLeftEvaluationMorphism( a, b ); <A morphism in Category of matrices over Q> gap> coev_ab := ClosedMonoidalLeftCoevaluationMorphism( a, b ); <A morphism in Category of matrices over Q> gap> coev_ba := ClosedMonoidalLeftCoevaluationMorphism( b, a ); <A morphism in Category of matrices over Q> gap> cocl_ev_ab := CoclosedMonoidalLeftEvaluationMorphism( a, b ); <A morphism in Category of matrices over Q> gap> cocl_ev_ba := CoclosedMonoidalLeftEvaluationMorphism( b, a ); <A morphism in Category of matrices over Q> gap> cocl_coev_ab := CoclosedMonoidalLeftCoevaluationMorphism( a, b ); <A morphism in Category of matrices over Q> gap> cocl_coev_ba := CoclosedMonoidalLeftCoevaluationMorphism( b, a ); <A morphism in Category of matrices over Q> gap> UnderlyingMatrix( ev_ab ) = TransposedMatrix( UnderlyingMatrix( cocl_ev_ab ) ); true gap> UnderlyingMatrix( coev_ab ) = TransposedMatrix( UnderlyingMatrix( cocl_coev_ab ) ); true gap> UnderlyingMatrix( coev_ba ) = TransposedMatrix( UnderlyingMatrix( cocl_coev_ba ) ); true gap> tensor_hom_adj_1_hom_ab := InternalHomToTensorProductLeftAdjunctMorphism( a, b, 1_hom_ab ); <A morphism in Category of matrices over Q> gap> cohom_tensor_adj_1_cohom_ab := InternalCoHomToTensorProductLeftAdjunctMorphism( a, b, 1_cohom_ab ); <A morphism in Category of matrices over Q> gap> tensor_hom_adj_1_ab := TensorProductToInternalHomLeftAdjunctMorphism( a, b, 1_ab ); <A morphism in Category of matrices over Q> gap> cohom_tensor_adj_1_ab := TensorProductToInternalCoHomLeftAdjunctMorphism( a, b, 1_ab ); <A morphism in Category of matrices over Q> gap> ev_ab = tensor_hom_adj_1_hom_ab; true gap> cocl_ev_ba = cohom_tensor_adj_1_cohom_ab; true gap> coev_ba = tensor_hom_adj_1_ab; true gap> cocl_coev_ba = cohom_tensor_adj_1_ab; true gap> c := VectorSpaceObject(2,Q); <A vector space object over Q of dimension 2> gap> d := VectorSpaceObject(1,Q); <A vector space object over Q of dimension 1> gap> pre_compose := MonoidalPreComposeMorphism( a, b, c ); <A morphism in Category of matrices over Q> gap> post_compose := MonoidalPostComposeMorphism( a, b, c ); <A morphism in Category of matrices over Q> gap> pre_cocompose := MonoidalPreCoComposeMorphism( c, b, a ); <A morphism in Category of matrices over Q> gap> post_cocompose := MonoidalPostCoComposeMorphism( c, b, a ); <A morphism in Category of matrices over Q> gap> UnderlyingMatrix( pre_compose ) = TransposedMatrix( UnderlyingMatrix( pre_cocompose ) ); true gap> UnderlyingMatrix( post_compose ) = TransposedMatrix( UnderlyingMatrix( post_cocompose ) ); true gap> tp_hom_comp := TensorProductInternalHomCompatibilityMorphism( [ a, b, c, d ] ); <A morphism in Category of matrices over Q> gap> cohom_tp_comp := InternalCoHomTensorProductCompatibilityMorphism( [ b, d, a, c ] ); <A morphism in Category of matrices over Q> gap> UnderlyingMatrix( tp_hom_comp ) = TransposedMatrix( UnderlyingMatrix( cohom_tp_comp ) ); true gap> lambda := LambdaIntroduction( alpha ); <A morphism in Category of matrices over Q> gap> lambda_elim := LambdaElimination( a, b, lambda ); <A morphism in Category of matrices over Q> gap> alpha = lambda_elim; true gap> alpha_op := VectorSpaceMorphism( b, TransposedMatrix( UnderlyingMatrix( alpha ) ), a ); <A morphism in Category of matrices over Q> gap> colambda := CoLambdaIntroduction( alpha_op ); <A morphism in Category of matrices over Q> gap> colambda_elim := CoLambdaElimination( b, a, colambda ); <A morphism in Category of matrices over Q> gap> alpha_op = colambda_elim; true gap> UnderlyingMatrix( lambda ) = TransposedMatrix( UnderlyingMatrix( colambda ) ); true gap> delta := PreCompose( colambda, lambda); <A morphism in Category of matrices over Q> gap> Display( TraceMap( delta ) ); [ [ 9 ] ] A morphism in Category of matrices over Q gap> Display( CoTraceMap( delta ) ); [ [ 9 ] ] A morphism in Category of matrices over Q gap> TraceMap( delta ) = CoTraceMap( delta ); true gap> RankMorphism( a ) = CoRankMorphism( a ); true
Split epi summand
gap> Q := HomalgFieldOfRationals();; gap> a := VectorSpaceObject( 3, Q );; gap> b := VectorSpaceObject( 4, Q );; gap> homalg_matrix := HomalgMatrix( [ [ 1, 0, 0, 0 ], > [ 0, 1, 0, -1 ], > [ -1, 0, 2, 1 ] ], 3, 4, Q );; gap> alpha := VectorSpaceMorphism( a, homalg_matrix, b );; gap> Display( SomeReductionBySplitEpiSummand( alpha ) ); (an empty 0 x 1 matrix) A morphism in Category of matrices over Q gap> Display( SomeReductionBySplitEpiSummand_MorphismFromInputRange( alpha ) ); [ [ 0 ], [ 1 ], [ -1/2 ], [ 1 ] ] A morphism in Category of matrices over Q gap> Display( SomeReductionBySplitEpiSummand_MorphismToInputRange( alpha ) ); [ [ 0, 1, 0, 0 ] ] A morphism in Category of matrices over Q
Kernel
gap> Q := HomalgFieldOfRationals(); Q gap> V := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> W := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W ); <A morphism in Category of matrices over Q> gap> k := KernelObject( alpha ); <A vector space object over Q of dimension 1> gap> T := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> tau := VectorSpaceMorphism( T, [ [ 2, 2 ], [ 2, 2 ] ], V ); <A morphism in Category of matrices over Q> gap> k_lift := KernelLift( alpha, tau ); <A morphism in Category of matrices over Q> gap> HasKernelEmbedding( alpha ); false gap> KernelEmbedding( alpha ); <A split monomorphism in Category of matrices over Q>
gap> Q := HomalgFieldOfRationals(); Q gap> V := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> W := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W ); <A morphism in Category of matrices over Q> gap> k := KernelObject( alpha ); <A vector space object over Q of dimension 1> gap> T := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> tau := VectorSpaceMorphism( T, [ [ 2, 2 ], [ 2, 2 ] ], V ); <A morphism in Category of matrices over Q> gap> k_lift := KernelLift( alpha, tau ); <A morphism in Category of matrices over Q> gap> HasKernelEmbedding( alpha ); false
gap> Q := HomalgFieldOfRationals(); Q gap> V := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> W := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> alpha := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W ); <A morphism in Category of matrices over Q> gap> k := KernelObject( alpha ); <A vector space object over Q of dimension 1> gap> k_emb := KernelEmbedding( alpha ); <A split monomorphism in Category of matrices over Q> gap> IsEqualForObjects( Source( k_emb ), k ); true gap> V := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> W := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> beta := VectorSpaceMorphism( V, [ [ 1, 1, 1 ], [ -1, -1, -1 ] ], W ); <A morphism in Category of matrices over Q> gap> k_emb := KernelEmbedding( beta ); <A split monomorphism in Category of matrices over Q> gap> IsIdenticalObj( Source( k_emb ), KernelObject( beta ) ); true
FiberProduct
gap> Q := HomalgFieldOfRationals(); Q gap> A := VectorSpaceObject( 1, Q ); <A vector space object over Q of dimension 1> gap> B := VectorSpaceObject( 2, Q ); <A vector space object over Q of dimension 2> gap> C := VectorSpaceObject( 3, Q ); <A vector space object over Q of dimension 3> gap> AtoC := VectorSpaceMorphism( A, [ [ 1, 2, 0 ] ], C ); <A morphism in Category of matrices over Q> gap> BtoC := VectorSpaceMorphism( B, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], C ); <A morphism in Category of matrices over Q> gap> P := FiberProduct( AtoC, BtoC ); <A vector space object over Q of dimension 1> gap> p1 := ProjectionInFactorOfFiberProduct( [ AtoC, BtoC ], 1 ); <A morphism in Category of matrices over Q> gap> p2 := ProjectionInFactorOfFiberProduct( [ AtoC, BtoC ], 2 ); <A morphism in Category of matrices over Q>