GAP (FreydCategoriesForCAP) - Chapter 16: Example on tensor products in Freyd categories

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16 Example on tensor products in Freyd categories

16.1 Tensor products for categories of rows

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

16.2 Tensor products for categories of columns

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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