Fragmentary structure of global knowledge: constructive processes for interoperability

The application of category theory to interoperability to increase understanding of the problems and to facilitate the development of practical tools for their solution.

Category theory is naturally suited to handling interoperability. The use of first order predicate logic in many information systems may be justified through its completeness. However, the work of Gödel shows that such systems are undecidable if they rely on formal systems of number and/or sets. For interoperability dyadic higher order logic is required, which is neither complete nor decidable if based on sets. However, pure category theory is still axiomatic so is also neither complete nor decidable. Applied category theory based on cartesian closed categories for process is natural and is both complete and decidable. Gödel's theorems therefore do not apply.

The paper finds that composed adjunctions appear particularly well‐suited for modelling interoperability, with composition of distinct functors for mapping across a number of levels and of endofunctors for business process interoperability. The monad/comonad category provides a powerful abstraction of the business process. The development of a tool based on categorial principles written in Haskell may be a way forward but only as an initial set model approach.

This paper applies categorial constructions which permit a natural formal approach to interoperability.