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9 Finite completion of a finitely presented (linear) category
 9.1 Constructors
 9.2 Attributes
 9.3 GAP Categories
 9.4 Examples

9 Finite completion of a finitely presented (linear) category

9.1 Constructors

9.1-1 FiniteCompletion
‣ FiniteCompletion( B )( operation )
‣ FiniteCompletion( B, H )( operation )

Returns: a CAP category

Construct a finite completion category.

9.2 Attributes

9.2-1 UnderlyingCategory
‣ UnderlyingCategory( finite_completion )( attribute )

9.2-2 EmbeddingOfUnderlyingCategory
‣ EmbeddingOfUnderlyingCategory( finite_completion )( attribute )

Returns: a CAP functor

9.3 GAP Categories

9.3-1 IsFiniteCompletion
‣ IsFiniteCompletion( category )( filter )

Returns: true or false

The GAP category of a finite completion category.

9.3-2 IsCellInFiniteCompletion
‣ IsCellInFiniteCompletion( cell )( filter )

Returns: true or false

The GAP category of cells in a finite completion category.

9.3-3 IsObjectInFiniteCompletion
‣ IsObjectInFiniteCompletion( obj )( filter )

Returns: true or false

The GAP category of objects in a finite completion category.

9.3-4 IsMorphismInFiniteCompletion
‣ IsMorphismInFiniteCompletion( mor )( filter )

Returns: true or false

The GAP category of morphisms in a finite completion category.

9.4 Examples

9.4-1 The terminal category as a finite completion of the initial category
gap> LoadPackage( "FunctorCategories", false );
true
gap> T := FiniteCompletion( InitialCategory( ) );
FiniteCompletion( InitialCategory( ) )
gap> IsIdenticalObj( RangeCategoryOfHomomorphismStructure( T ), T );
true
gap> IsTerminalCategory( T );
true
gap> Display( T );
A CAP category with name FiniteCompletion( InitialCategory( ) ):

110 primitive operations were used to derive 627 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
* IsLeftClosedMonoidalCategory
* IsLeftCoclosedMonoidalCategory
* IsAbelianCategoryWithEnoughInjectives
* IsAbelianCategoryWithEnoughProjectives
* IsSymmetricClosedMonoidalLattice
* IsSymmetricCoclosedMonoidalLattice
* IsBooleanAlgebra
* IsRigidSymmetricClosedMonoidalCategory
* IsRigidSymmetricCoclosedMonoidalCategory
and furthermore mathematically
* IsDiscreteCategory
* IsFinitelyPresentedLinearCategory
* IsLinearClosureOfACategory
* IsLocallyOfFiniteInjectiveDimension
* IsLocallyOfFiniteProjectiveDimension
* IsTerminalCategory
* IsTotalOrderCategory
gap> i := InitialObject( T );
<An object in FiniteCompletion( InitialCategory( ) )>
gap> t := TerminalObject( T );
<An object in FiniteCompletion( InitialCategory( ) )>
gap> z := ZeroObject( T );
<A zero object in FiniteCompletion( InitialCategory( ) )>
gap> Display( i );
An object in CoPreSheaves( InitialCategory( ), InitialCategory( ) )
given by the above data

An object in FiniteCompletion( InitialCategory( ) ) given by the above data
gap> Display( t );
An object in CoPreSheaves( InitialCategory( ), InitialCategory( ) )
given by the above data

An object in FiniteCompletion( InitialCategory( ) ) given by the above data
gap> Display( z );
An object in CoPreSheaves( InitialCategory( ), InitialCategory( ) )
given by the above data

An object in FiniteCompletion( InitialCategory( ) ) given by the above data
gap> IsIdenticalObj( i, z );
false
gap> IsIdenticalObj( t, z );
false
gap> IsEqualForObjects( i, z );
true
gap> IsEqualForObjects( t, z );
true
gap> IsWellDefined( z );
true
gap> id_z := IdentityMorphism( z );
<A zero, identity morphism in FiniteCompletion( InitialCategory( ) )>
gap> fn_z := ZeroObjectFunctorial( T );
<A zero, isomorphism in FiniteCompletion( InitialCategory( ) )>
gap> IsWellDefined( fn_z );
true
gap> IsEqualForMorphisms( id_z, fn_z );
true
gap> IsCongruentForMorphisms( id_z, fn_z );
true
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