Final coalgebras in accessible categories

Abstract

We propose a construction of the final coalgebra for a finitary endofunctor of a finitely accessible category and study conditions under which this construction is available. Our conditions always apply when the accessible category is cocomplete, and is thus a locally finitely presentable (l.f.p.) category, and we give an explicit and uniform construction of the final coalgebra in this case. On the other hand, our results also apply to some interesting examples of final coalgebras beyond the realm of l.f.p. categories. In particular, we construct the final coalgebra for every finitary endofunctor on the category of linear orders, and analyse Freyd's coalgebraic characterisation of the closed unit as an instance of this construction. We use and extend results of Tom Leinster, developed for his study of self-similar objects in topology, relying heavily on his formalism of modules (corresponding to endofunctors) and complexes for a module.