Terminal Coalgebras in Countably Many Steps

Abstract

We present a collection of results that imply that an endofunctor on a category has a terminal coalgebra obtainable as a countable limit of its terminal-coalgebra chain. This holds for finitary endofunctors on locally finitely presentable categories under conditions on both the functor and the category. We 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 yields an endofunctor $\mathscr{V}$ on the category of Hausdorff spaces. Vietoris polynomial endofunctors are built from $ \mathscr{V}$, the identity and constant functors by forming products, coproducts and compositions. Their terminal coalgebras are obtained in $\omega$ steps. We then turn to the class of Hausdorff polynomial functors on the category of metric spaces, which is analogous but uses in lieu of $\mathscr{V}$ the Hausdorff functor $\mathcal{H}$. We prove they have terminal coalgebras obtained in $\omega + \omega$ steps. Finally, 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.