How is it possible to integrate databases that have ontologies written in incompatible languages, with different schemas, and with different data models?

Can we give formal definitions for ontology language, ontology integration, and ontology mapping?

1. KIF, Ontolingua, Flora, RDF, etc. are formal logics.
    
2. An ontolgy is a theory in a formal logic.
    
3. But what is a logic?
    
4. And how could we hope to translate between theories written in different logics?
    
5. Category theory provides ideas and techniques at the right level of abstraction (which is very high!).
    
6. The theory of institutions formalizes the notion of logic. Abstraction of the model theory and metamathematics of Tarski etc.
    
7. Institution morphisms formalizes translations between logics.
    
8. Very very abstract, but is being used in the European CoFI project.
    
9. Razvan Diaconescu has an elegant way to integrate theories (i.e. ontologies) in different logics using Grothendieck institutions.

More information on some of these topics can be found on the SEEK project page.