Terminal Coalgebras in Countably Many Steps

We present a collection of results that imply that an endofunctor on a category has a terminal object obtainable as a countable limit of its terminal-coalgebra chain. This holds for finitary endofunctors preserving nonempty binary intersections on locally finitely presentable categories, assuming that the posets of strong quotients and subobjects of finitely presentable objects satisfy the descending chain condition. This allows one to adapt finiteness arguments that were originally advanced by Worrell concerning terminal coalgebras for finitary set functors. Examples include the categories of sets, posets, vector spaces, graphs, nominal sets, and presheaves on finite sets. Worrell also described, without proof, the terminal-coalgebra chain of the finite power-set functor. We provide a detailed proof following his ideas. We then turn to polynomial endofunctors on the categories of Hausdorff topological spaces and metric spaces. The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor $\mathscr{V}$ on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from $\mathscr{V}$, the identity and constant functors by forming products, coproducts and compositions. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of $\mathscr{V}$ the Hausdorff functor $\mathcal{H}$. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. We show that every finitary endofunctor on the category of vector spaces over a fixed field again has a terminal coalgebra obtained in $\omega+\omega$ steps. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes $\omega$ steps, one needs $\omega + \omega$ steps in general for our other concrete settings.