Goto Chapter: Top 1 2 3 4 Ind

### 4 The finitely Z-graded closure of a category

#### 4.1 Constructors

 ‣ FinitelyZGradedClosureCategory( C ) ( attribute )

Returns: a CAP category

Construct the negatively Z-graded closure of the category C.

gap> LoadPackage( "GradedCategories" );
true
true
gap> Q := HomalgFieldOfRationals( );
Q
gap> Qmat := MatrixCategory( Q );
Category of matrices over Q
gap> ZQmat := FinitelyZGradedClosureCategory( Qmat );
FinitelyZGradedClosureCategory( Category of matrices over Q )
gap> z := ZeroObject( ZQmat );
<A zero object in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> Sublist( z, [ -1 .. 2 ] );
[ <A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0> ]
gap> o0 := ZQmat[0];
<An object in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> Sublist( o0, [ -1 .. 2 ] );
[ <A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 1>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0> ]
gap> IsZero( o0 );
false
gap> o1 := ZQmat[1];
<An object in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> Sublist( o1, [ -1 .. 2 ] );
[ <A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 1>,
<A vector space object over Q of dimension 0> ]
gap> o00 := DirectSum( o0, o0 );
<An object in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> Sublist( o00, [ -1 .. 2 ] );
[ <A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 2>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0> ]
gap> o0 = TensorUnit( ZQmat );
true
gap> TensorProduct( o1, ZQmat[-1] ) = o0;
true
gap> o11 := TensorProduct( o00, o1 );
<An object in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> Sublist( o11, [ -1 .. 2 ] );
[ <A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 0>,
<A vector space object over Q of dimension 2>,
<A vector space object over Q of dimension 0> ]
gap> IsZero( TensorProduct( z, o11 ) );
true
gap> lu := LeftUnitor( o00 );
<A morphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> IsIsomorphism( lu );
true
gap> lu;
<An isomorphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> slu := Sublist( lu, [ -1 .. 2 ] );;
gap> List( slu, IsIsomorphism );
[ true, true, true, true ]
gap> List( slu, IsZero );
[ true, false, true, true ]
gap> slu;
[ <A zero, isomorphism in Category of matrices over Q>,
<An isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q> ]
gap> Display( lu[0] );
[ [  1,  0 ],
[  0,  1 ] ]

An isomorphism in Category of matrices over Q
gap> ru := RightUnitor( o00 );
<A morphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> IsIsomorphism( ru );
true
gap> ru;
<An isomorphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> sru := Sublist( ru, [ -1 .. 2 ] );;
gap> List( sru, IsIsomorphism );
[ true, true, true, true ]
gap> List( sru, IsZero );
[ true, false, true, true ]
gap> sru;
[ <A zero, isomorphism in Category of matrices over Q>,
<An isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q> ]
gap> Display( ru[0] );
[ [  1,  0 ],
[  0,  1 ] ]

An isomorphism in Category of matrices over Q
gap> lr := AssociatorLeftToRight( o0, o1, o00 );
<A morphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> IsIsomorphism( lr );
true
gap> lr;
<An isomorphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> slr := Sublist( lr, [ -1 .. 2 ] );;
gap> List( slr, IsIsomorphism );
[ true, true, true, true ]
gap> List( slr, IsZero );
[ true, true, false, true ]
gap> slr;
[ <A zero, isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q>,
<An isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q> ]
gap> Display( lr[1] );
[ [  1,  0 ],
[  0,  1 ] ]

An isomorphism in Category of matrices over Q
gap> rl := AssociatorRightToLeft( o0, o1, o00 );
<A morphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> IsIsomorphism( rl );
true
gap> rl;
<An isomorphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> srl := Sublist( rl, [ -1 .. 2 ] );;
gap> List( srl, IsIsomorphism );
[ true, true, true, true ]
gap> List( srl, IsZero );
[ true, true, false, true ]
gap> srl;
[ <A zero, isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q>,
<An isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q> ]
gap> Display( rl[1] );
[ [  1,  0 ],
[  0,  1 ] ]

An isomorphism in Category of matrices over Q
gap> b := Braiding( o11, o00 );
<A morphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> IsIsomorphism( b );
true
gap> b;
<An isomorphism in
FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> sb := Sublist( b, [ -1 .. 2 ] );;
gap> List( sb, IsIsomorphism );
[ true, true, true, true ]
gap> List( sb, IsZero );
[ true, true, false, true ]
gap> sb;
[ <A zero, isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q>,
<An isomorphism in Category of matrices over Q>,
<A zero, isomorphism in Category of matrices over Q> ]
gap> Display( b[1] );
[ [  1,  0,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  0,  1 ] ]

An isomorphism in Category of matrices over Q


#### 4.2 GAP Categories

 ‣ IsFinitelyZGradedClosureCategory( object ) ( filter )

Returns: true or false

The GAP category of finitely Z-graded categories.

 ‣ IsCellInFinitelyZGradedClosureCategory( object ) ( filter )

Returns: true or false

The GAP category of cells in a finitely Z-graded category.

 ‣ IsObjectInFinitelyZGradedClosureCategory( object ) ( filter )

Returns: true or false

The GAP category of objects in a finitely Z-graded category.

 ‣ IsMorphismInFinitelyZGradedClosureCategory( morphism ) ( filter )
Returns: true or false