# Formalizing Ontology Alignment and its Operations with Category Theory

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Antoine Zimmermann, Markus Krötzsch, Jérôme Euzenat, Pascal Hitzler

Formalizing Ontology Alignment and its Operations with Category Theory

**Abstract.** An ontology alignment is the expression of relations between different
ontologies. In order to view alignments independently from the language expressing ontologies and from the techniques used for finding the alignments, we use a
category-theoretical model in which ontologies are the objects. We introduce a categorical structure, called V-alignment, made of a pair of morphisms with a common domain having the ontologies as codomain. This structure serves to design an algebra that describes formally what are ontology merging, alignment composition, union and intersection using categorical constructions. This enables combining alignments of various provenance. Although the desirable properties of this
algebra make such abstract manipulation of V-alignments very simple, it is practically not well fitted for expressing complex alignments: expressing subsumption
between entities of two different ontologies demands the deﬁnition of non-standard
categories of ontologies. We consider two approaches to solve this problem. The
first one extends the notion of V-alignments to a more complex structure called
W-alignments: a formalization of alignments relying on “bridge axioms.” The second one relies on an elaborate concrete category of ontologies that offers high expressive power. We show that these two extensions have different advantages that
may be exploited in different contexts (viz., merging, composing, joining or meeting): the first one efficiently processes ontology merging thanks to the possible use
of categorical institution theory, while the second one benefits from the simplicity
of the algebra of V-alignments.

*Published at FOIS2006 (Conference paper) *

*Download PDF* *(last update: November 1 2006)*

## Citation details

- Antoine Zimmermann, Markus Krötzsch, Jérôme Euzenat, Pascal Hitzler. Formalizing Ontology Alignment and its Operations with Category Theory. In Brandon Bennett, Christiane Fellbaum, eds.: Proceedings of the Fourth International Conference on Formal Ontology in Information Systems (FOIS-06), pp. 277–288. IOS Press 2006.