‣ FiniteCompletion( B ) | ( operation ) |
‣ FiniteCompletion( B, H ) | ( operation ) |
Returns: a CAP category
Construct a finite completion category.
‣ UnderlyingCategory( finite_completion ) | ( attribute ) |
‣ CoYonedaEmbeddingOfUnderlyingCategory( finite_completion ) | ( attribute ) |
Returns: a CAP functor
‣ IsFiniteCompletion( category ) | ( filter ) |
Returns: true or false
The GAP category of a finite completion category.
‣ IsCellInFiniteCompletion( cell ) | ( filter ) |
Returns: true or false
The GAP category of cells in a finite completion category.
‣ IsObjectInFiniteCompletion( obj ) | ( filter ) |
Returns: true or false
The GAP category of objects in a finite completion category.
‣ IsMorphismInFiniteCompletion( mor ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a finite completion category.
gap> LoadPackage( "FunctorCategories", false ); true gap> T := FiniteCompletion( InitialCategory( ) ); FiniteCompletion( InitialCategory( ) ) gap> IsTerminalCategory( T ); true gap> Display( T ); A CAP category with name FiniteCompletion( InitialCategory( ) ): 91 primitive operations were used to derive 482 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsLinearCategoryOverCommutativeRing * IsBicartesianClosedCategory * IsBicartesianCoclosedCategory * IsAbelianCategoryWithEnoughInjectives * IsAbelianCategoryWithEnoughProjectives * IsRigidSymmetricClosedMonoidalCategory * IsRigidSymmetricCoclosedMonoidalCategory and furthermore mathematically * IsFinitelyPresentedCategory * IsFinitelyPresentedLinearCategory * IsLinearClosureOfACategory * IsLocallyOfFiniteInjectiveDimension * IsLocallyOfFiniteProjectiveDimension * IsSkeletalCategory * IsStrictCartesianCategory * IsStrictCocartesianCategory * IsStrictMonoidalCategory * IsTerminalCategory 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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