Coalgebra morphisms subsume open maps

Abstract

We relate two abstract notions of bisimulation, induced by open maps and by coalgebras morphisms, respectively.We show that open maps correspond to coalgebra morphisms for a suitable chosen endofunctor in a category of many sorted sets. This demonstrates that the notion of open-maps bisimilarity is of essentially coalgebraic nature. We find moreover characterizations of subcategories of coalgebras corresponding to some relevant model categories: category of presheaves is isomorphic to the subcategory of consistent coalgebras with lax cohomomorphisms; a suitably chosen subcategory of many sorted algebras is equivalent to the full subcategory of presheaves preserving finite products.