Strongly Complete Logics for Coalgebras

Coalgebras for a functor T on a category X model many different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary strongly complete specification languages for Set-based coalgebras. We show how to associate a finitary logic to any finite-sets preserving functor T and prove the logic to be strongly complete under a mild condition on T . The proof is based on the following result. An endofunctor on a variety has a presentation by operations and equations iff it preserves sifted colimits.