🤖 AI Summary
This paper investigates the existence and computational complexity of terminal coalgebras for endofunctors on locally finitely presentable categories. For finite endofunctors preserving nonempty binary intersections, we prove that a terminal coalgebra exists—and is constructible as a countable limit of a terminal-coalgebra chain—provided the poset of strong quotients (or subobjects) of finitely generated objects satisfies the descending chain condition. We then systematically construct Hausdorff-type polynomial functors on metric spaces, establishing for the first time that their terminal coalgebras require exactly ω+ω steps to converge—strictly exceeding the ω-step convergence of Vietoris-type functors—and thereby precisely characterizing their ordinal complexity gap. Additionally, we provide a novel proof that the closed-set functor admits no fixed point in the category of metric spaces. Collectively, these results unify and establish the existence of both initial algebras and terminal coalgebras for Vietoris- and Hausdorff-type polynomial functors across topological and metric space categories.
📝 Abstract
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. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. 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. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes $omega$ steps, one needs $omega + omega$ steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.
