gap> R := HomalgFieldOfRationalsInSingular() * "a,b,c,d,e,f,g,h,i,j";; gap> C := CategoryOfRows( R );; gap> T := TensorUnit( C );; gap> IsWellDefined( T ); true
We test the naturality of the braiding.
gap> R2 := DirectSum( T, T );; gap> R3 := DirectSum( T, R2 );; gap> R4 := DirectSum( R2, R2 );; gap> alpha := CategoryOfRowsMorphism( T, HomalgMatrix( "[ a, b, c, d ]", 1, 4, R ), R4 );; gap> beta := CategoryOfRowsMorphism( R2, HomalgMatrix( "[ e, f, g, h, i, j ]", 2, 3, R ), R3 );; gap> IsCongruentForMorphisms( > PreCompose( Braiding( T, R2 ), TensorProductOnMorphisms( beta, alpha ) ), > PreCompose( TensorProductOnMorphisms( alpha, beta ), Braiding( R4, R3 ) ) > ); true
We compute the torsion part of a f.p. module with the help of the induced tensor structure on the Freyd category.
gap> M := FreydCategoryObject( alpha );; gap> mu := MorphismToBidual( M );; gap> co := CoastrictionToImage( mu );; gap> IsIsomorphism( co ); true
gap> R := HomalgFieldOfRationalsInSingular() * "a,b,c,d,e,f,g,h,i,j";; gap> C := CategoryOfColumns( R );; gap> T := TensorUnit( C );; gap> IsWellDefined( T ); true
We test the naturality of the braiding.
gap> R2 := DirectSum( T, T );; gap> R3 := DirectSum( T, R2 );; gap> R4 := DirectSum( R2, R2 );; gap> alpha := CategoryOfColumnsMorphism( T, HomalgMatrix( "[ a, b, c, d ]", 4, 1, R ), R4 );; gap> beta := CategoryOfColumnsMorphism( R2, HomalgMatrix( "[ e, f, g, h, i, j ]", 3, 2, R ), R3 );; gap> IsCongruentForMorphisms( > PreCompose( Braiding( T, R2 ), TensorProductOnMorphisms( beta, alpha ) ), > PreCompose( TensorProductOnMorphisms( alpha, beta ), Braiding( R4, R3 ) ) > ); true
We compute the torsion part of a f.p. module with the help of the induced tensor structure on the Freyd category.
gap> M := FreydCategoryObject( alpha );; gap> mu := MorphismToBidual( M );; gap> co := CoastrictionToImage( mu );; gap> IsIsomorphism( co ); true
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