This paper studies coalgebras from the perspective of the finitary observations that can be made of their behaviours. Based on the terminal sequence, notions of finitary behaviours and finitary predicates are introduced. A category Behω(T) of coalgebras with morphisms preserving finitary behaviours is defined. We then investigate definability and compactness for finitary coalgebraic modal logic, show that the final object in Behω(T) generalises the notion of a canonical model in modal logic, and study the topology induced on a coalgebra by the finitary part of the terminal sequence.
A. Kurz and D. Pattinson, “Definability, Canonical Models, Compactness for Finitary Coalgebraic Modal Logic,” Electronic Notes in Theoretical Computer Science, vol. 65, no. 1, pp. 135–155, Oct. 2002. DOI: 10.1016/S1571-0661(04)80363-3
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