Converting an Ontology into a Category

Category Theory is the mathematical theory of structure. We are interested in studying the practical aspects of categorical completion of ontologies, a process of transforming ontologies into categories. This process is not generic, (for reasons not explained here), apart from the main axioms a category must satisfy, we require the presence of other elements such as all colimits and a terminal object.

The first thing to do is to represent the identity morphism that all objects in category must have. This can be done simply by adding a reflexive and transitive object property hasIdentityMorphism with range Thing and domain Thing, so that all things have such identity property.

A second task to accomplish is to endow an ontology O with a terminal object class to represent the terminal object in the categorical completion C of O. I took the Pizza ontology and added a subclass TerminalObject to Thing. Also, in order to ensure "all things" have a terminal object, a transitive object property (terminalProperty) was added.

I use the Java OWLAPI to materialize these operations. These are the relevant lines of code for the two first steps explained above:

OWLOntologyManager manager = OWLManager.createOWLOntologyManager();
IRI iri = IRI.create("http://www.co-ode.org/ontologies/pizza/pizza.owl");
OWLOntology pizzaOntology = manager.loadOntologyFromOntologyDocument(iri);

OWLClass Thing = factory.getOWLClass(IRI.create("http://www.w3.org/2002/07/owl#Thing"));

OWLClass TerminalObject = factory.getOWLClass(IRI.create(iri + "#TerminalObject"));
OWLObjectProperty terminalProperty = factory.getOWLObjectProperty(IRI.create(iri + "#hasTerminalObject"));
        
OWLTransitiveObjectPropertyAxiom TerminalObjectProperty = factory.getOWLTransitiveObjectPropertyAxiom(terminalProperty);
manager.addAxiom(pizzaOntology, TerminalObjectProperty);
Set terminalAxioms = new HashSet();
        
terminalAxioms.add(factory.getOWLObjectPropertyDomainAxiom(terminalProperty, Thing));
terminalAxioms.add(factory.getOWLObjectPropertyRangeAxiom(terminalProperty, TerminalObject));
terminalAxioms.add(factory.getOWLSubClassOfAxiom(TerminalObject, Thing));
        
manager.addAxioms(pizzaOntology, terminalAxioms);

File file = new File("/tmp/catcomp_pizza.owl");
manager.saveOntology(pizzaOntology, owlxmlFormat, IRI.create(file.toURI()));

This is screenshot taken from a Protege/OntoGraf view on the new transformed ontology.