‣ UnderlyingCategory( DC ) | ( attribute ) |
Return the category \(C\) underlying the distributive closure with strict products and coproducts category DC := FreeDistributiveCategoryWithStrictProductAndCoproducts( \(C\) ).
‣ FreeDistributiveCategoryWithStrictProductAndCoproducts( C ) | ( attribute ) |
Returns: a CAP category
Construct the free distributive closure category with strict products and coproducts of a category C.
gap> LoadPackage( "FiniteCocompletions" ); true gap> LoadPackage( "Algebroids" ); true gap> Q := RightQuiver( "Q(a,b,c)[]" ); Q(a,b,c)[] gap> C := FreeCategory( Q ); FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) gap> DC := FreeDistributiveCategoryWithStrictProductAndCoproducts( C ); FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) gap> Display( DC ); A CAP category with name FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ): 30 primitive operations were used to derive 139 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsEquippedWithHomomorphismStructure * IsDistributiveCategory and furthermore mathematically * IsStrictCartesianCategory * IsStrictCocartesianCategory gap> a := DC.a; <An object in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> b := DC.b; <An object in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> c := DC.c; <An object in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> DirectProduct( DirectProduct( a, b ), c ) = > DirectProduct( a, DirectProduct( b, c ) ); true gap> axb_u_axc := Coproduct( DirectProduct( a, b ), DirectProduct( a, c ) ); <An object in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> a_x_buc := DirectProduct( a, Coproduct( b, c ) ); <An object in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> axb_u_axc = a_x_buc; true gap> HomStructure( axb_u_axc, a_x_buc ); |1| gap> hom := MorphismsOfExternalHom( a_x_buc, axb_u_axc ); [ <A morphism in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> ] gap> phi := hom[1]; <A morphism in FreeDistributiveCategoryWithStrictProductAndCoproducts( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> gap> IsOne( phi ); true gap> Source( phi ) = axb_u_axc; true gap> Target( phi ) = a_x_buc; true gap> IsOne( phi ); true gap> phi = LeftCartesianDistributivityFactoring( a, [ b, c ] ); true gap> Id := ExtendFunctorToFreeDistributiveCategoryWithStrictProductAndCoproducts( > EmbeddingOfUnderlyingCategory( DC ) ); Extension to FreeDistributiveCategoryWithStrictProductAndCoproducts( Source( Embedding functor into a free distributive closure category with strict products and coproducts ) ) gap> Id( phi ) = phi; true
‣ EmbeddingOfUnderlyingCategory( UC ) | ( attribute ) |
Returns: a CAP functor
The full embedding functor from the category \(C\) underlying the free distributive closure category DC with strict products and coproducts into DC.
‣ ExtendFunctorToFreeDistributiveCategoryWithStrictProductAndCoproducts( DC ) | ( attribute ) |
Returns: a CAP functor
The full embedding functor from the category \(C\) underlying the free distributive closure category DC with strict products and coproducts into DC.
‣ IsFreeDistributiveCategoryWithStrictProductAndCoproducts( arg ) | ( filter ) |
Returns: true or false
The GAP category of free distributive closure categories with strict products and coproducts of a category.
‣ IsCellInFreeDistributiveCategoryWithStrictProductAndCoproducts( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in the free distributive closure category with strict products and coproducts of a category.
‣ IsObjectInFreeDistributiveCategoryWithStrictProductAndCoproducts( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in the free distributive closure category with strict products and coproducts of a category.
‣ IsMorphismInFreeDistributiveCategoryWithStrictProductAndCoproducts( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the free distributive closure category with strict products and coproducts of a category.
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