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3 Free distributive closure category with strict products and coproducts
 3.1 Attributes
 3.2 Constructors
 3.3 GAP categories

3 Free distributive closure category with strict products and coproducts

3.1 Attributes

3.1-1 UnderlyingCategory
‣ UnderlyingCategory( DC )( attribute )

Return the category C underlying the distributive closure with strict products and coproducts category DC := FreeDistributiveCategoryWithStrictProductAndCoproducts( C ).

3.2 Constructors

3.2-1 FreeDistributiveCategoryWithStrictProductAndCoproducts
‣ 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 137 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

3.2-2 EmbeddingOfUnderlyingCategory
‣ 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.

3.2-3 ExtendFunctorToFreeDistributiveCategoryWithStrictProductAndCoproducts
‣ 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.

3.3 GAP categories

3.3-1 IsFreeDistributiveCategoryWithStrictProductAndCoproducts
‣ IsFreeDistributiveCategoryWithStrictProductAndCoproducts( arg )( filter )

Returns: true or false

The GAP category of free distributive closure categories with strict products and coproducts of a category.

3.3-2 IsCellInFreeDistributiveCategoryWithStrictProductAndCoproducts
‣ 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.

3.3-3 IsObjectInFreeDistributiveCategoryWithStrictProductAndCoproducts
‣ 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.

3.3-4 IsMorphismInFreeDistributiveCategoryWithStrictProductAndCoproducts
‣ 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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