Goto Chapter: Top 1 2 Ind

### 1 The standard category

#### 1.1 Attributes

##### 1.1-1 UnderlyingMatrixGroup
 ‣ UnderlyingMatrixGroup( arg ) ( attribute )

##### 1.1-2 UnderlyingRing
 ‣ UnderlyingRing( arg ) ( attribute )

##### 1.1-3 UnderlyingTwistingRingMap
 ‣ UnderlyingTwistingRingMap( arg ) ( attribute )

##### 1.1-4 MatrixGroupElement
 ‣ MatrixGroupElement( arg ) ( attribute )

##### 1.1-5 UnderlyingRingElement
 ‣ UnderlyingRingElement( arg ) ( attribute )

#### 1.2 Constructors

##### 1.2-1 StandardCategory
 ‣ StandardCategory( W, k ) ( operation )

Return the standard category over the commutative base ring k.

gap> LoadPackage( "HeckeCategories" );
true
gap> W := Group( [[0,1,0],[1,0,0],[0,0,1]], [[1,0,0],[0,0,1],[0,1,0]] );
Group([ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ],
[ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ] ] ])
gap> k := HomalgFieldOfRationalsInSingular( );
Q
gap> Std := StandardCategory( W, k );
StandardCategory( W, Q )
gap> Display( Std );
A CAP category with name StandardCategory( W, Q ):

20 primitive operations were used to derive 81 operations for this category
which algorithmically
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
* IsMonoidalCategory
and furthermore mathematically
* IsStrictMonoidalCategory
gap> R := UnderlyingRing( Std );
Q[a1,a2,a3]
gap> ExportVariables( R );
[ a1, a2, a3 ]
gap> x := [[0,1,0],[1,0,0],[0,0,1]];
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ]
gap> rx := x / Std;
<An object in StandardCategory( W, Q )>
gap> IsWellDefined( rx );
true
gap> Display( rx );
[ [ 0, 1, 0 ],
[ 1, 0, 0 ],
[ 0, 0, 1 ] ]
gap> y := [[1,0,0],[0,0,1],[0,1,0]];
[ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ] ]
gap> ry := y / Std;
<An object in StandardCategory( W, Q )>
gap> rx = ry;
false
gap> z := [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 2 ] ];
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 2 ] ]
gap> IsWellDefined( z / Std );
false
gap> rxy := TensorProduct( rx, ry );
<An object in StandardCategory( W, Q )>
gap> Display( rxy );
[ [  0,  0,  1 ],
[  1,  0,  0 ],
[  0,  1,  0 ] ]
gap> id_x := IdentityMorphism( rx );
<An identity morphism in StandardCategory( W, Q )>
gap> Display( id_x );
1
gap> phi_x := MorphismConstructor( rx, a1 * a2, rx );
<A morphism in StandardCategory( W, Q )>
gap> IsWellDefined( phi_x );
true
gap> Display( phi_x );
a1*a2
gap> IsOne( phi_x );
false
gap> Display( 2 * a1 * phi_x );
2*a1^2*a2
gap> Display( -phi_x );
-a1*a2
gap> psi_x := MorphismConstructor( rx, a2 * a1, rx );
<A morphism in StandardCategory( W, Q )>
gap> phi_x = psi_x;
true
gap> Display( phi_x - psi_x );
0
gap> Display( PreCompose( phi_x, psi_x ) );
a1^2*a2^2
gap> phi_y := MorphismConstructor( ry, a2 * a3, ry );
<A morphism in StandardCategory( W, Q )>
gap> phi_x = phi_y;
false
gap> zero_xy := ZeroMorphism( rx, ry );
<A zero morphism in StandardCategory( W, Q )>
gap> Display( zero_xy );
0
gap> zeta_xy := MorphismConstructor( rx, a1*a2, ry );
<A morphism in StandardCategory( W, Q )>
gap> IsZero( zeta_xy );
true
gap> IsEqualForMorphisms( zero_xy, zeta_xy );
true
gap> IsCongruentForMorphisms( zero_xy, zeta_xy );
true
gap> eta_x := a3 * phi_x;
<A morphism in StandardCategory( W, Q )>
gap> Display( eta_x );
a1*a2*a3
gap> end_x := BasisOfExternalHom( rx, rx );
[ <An identity morphism in StandardCategory( W, Q )> ]
gap> Display( end_x[1] );
1
gap> CoefficientsOfMorphism( eta_x );
[ a1*a2*a3 ]
gap> BasisOfExternalHom( rx, ry );
[ ]
gap> CoefficientsOfMorphism( zeta_xy );
[ ]
gap> I := TensorUnit( Std );
<An object in StandardCategory( W, Q )>
gap> Display( I );
[ [ 1, 0, 0 ],
[ 0, 1, 0 ],
[ 0, 0, 1 ] ]
gap> chi_xy := TensorProduct( phi_x, phi_y );
<A morphism in StandardCategory( W, Q )>
gap> Display( chi_xy );
a1^2*a2*a3


#### 1.3 GAP categories

##### 1.3-1 IsStandardCategory
 ‣ IsStandardCategory( arg ) ( filter )

Returns: true or false

The GAP category of standard categories.

##### 1.3-2 IsCellInStandardCategory
 ‣ IsCellInStandardCategory( arg ) ( filter )

Returns: true or false

The GAP category of cells in a standard category.

##### 1.3-3 IsObjectInStandardCategory
 ‣ IsObjectInStandardCategory( arg ) ( filter )

Returns: true or false

The GAP category of objects in a standard category.

##### 1.3-4 IsMorphismInStandardCategory
 ‣ IsMorphismInStandardCategory( arg ) ( filter )

Returns: true or false

The GAP category of morphisms in a standard category.

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