gap> LoadPackage( "FunctorCategories", false, ">= 2024.03-10" );
true
gap> IntervalCategory;
IntervalCategory
gap> Display( IntervalCategory );
A CAP category with name IntervalCategory:

21 primitive operations were used to derive 329 operations for this category
 which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsEquippedWithHomomorphismStructure
* IsBooleanAlgebra
gap> SetOfObjects( IntervalCategory );
[ <(⊥)>, <(⊤)> ]
gap> SetOfGeneratingMorphisms( IntervalCategory );
[ (⊥)-[(⇒)]->(⊤) ]
gap> PSh := PreSheaves( IntervalCategory );
PreSheaves( IntervalCategory, IntervalCategory )
gap> Display( PSh );
A CAP category with name PreSheaves( IntervalCategory, IntervalCategory ):

56 primitive operations were used to derive 286 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsEquippedWithHomomorphismStructure
* IsHeytingAlgebra
and not yet algorithmically
* IsBiHeytingAlgebra
gap> Y := YonedaEmbeddingOfSourceCategory( PSh );
Yoneda embedding functor
gap> t := Y( TerminalObject( IntervalCategory ) );
<A projective object in PreSheaves( IntervalCategory, IntervalCategory )>
gap> IsTerminal( t );
true
gap> Display( t );
Image of <(⊥)>:
<(⊤)>

Image of <(⊤)>:
<(⊤)>

Image of (⊥)-[(⇒)]->(⊤):
(⊤)-[(⊤)]->(⊤)

An object in PreSheaves( IntervalCategory, IntervalCategory )
given by the above data
gap> f := Y( InitialObject( IntervalCategory ) );
<A projective object in PreSheaves( IntervalCategory, IntervalCategory )>
gap> IsInitial( f );
false
gap> Display( f );
Image of <(⊥)>:
<(⊤)>

Image of <(⊤)>:
<(⊥)>

Image of (⊥)-[(⇒)]->(⊤):
(⊥)-[(⇒)]->(⊤)

An object in PreSheaves( IntervalCategory, IntervalCategory )
given by the above data
gap> i := InitialObject( PSh );
<An object in PreSheaves( IntervalCategory, IntervalCategory )>
gap> Display( i );
Image of <(⊥)>:
<(⊥)>

Image of <(⊤)>:
<(⊥)>

Image of (⊥)-[(⇒)]->(⊤):
(⊥)-[(⊥)]->(⊥)

An object in PreSheaves( IntervalCategory, IntervalCategory )
given by the above data
gap> gens := SetOfGeneratingMorphisms( PSh );
[ <A morphism in PreSheaves( IntervalCategory, IntervalCategory )>,
  <A morphism in PreSheaves( IntervalCategory, IntervalCategory )> ]
gap> Display( gens[1] );
Image of <(⊥)>:
(⊥)-[(⇒)]->(⊤)

Image of <(⊤)>:
(⊥)-[(⊥)]->(⊥)

A morphism in PreSheaves( IntervalCategory, IntervalCategory )
given by the above data
gap> Display( gens[2] );
Image of <(⊥)>:
(⊤)-[(⊤)]->(⊤)

Image of <(⊤)>:
(⊥)-[(⇒)]->(⊤)

A morphism in PreSheaves( IntervalCategory, IntervalCategory )
given by the above data
gap> coYgens := List( gens, CoYonedaLemmaOnMorphisms );
[ <A morphism in
   FiniteColimitCompletionWithStrictCoproducts( IntervalCategory )>,
  <A morphism in
   FiniteColimitCompletionWithStrictCoproducts( IntervalCategory )> ]
gap> List( coYgens, IsWellDefined );
[ true, true ]