Presenting Point-Glued Self-Similar Objects as Final Coalgebras in Metric Spaces via Completion: A Categorical Hutchinson Theorem

Abstract

We study self-similar spaces obtained by repeatedly gluing finitely many scaled copies of a pointed metric space along prescribed boundary points. A finite gluing specification defines endofunctors on suitable categories of pointed sets and pointed metric spaces. Under non-degeneracy and metric admissibility assumptions, the initial algebra of the set functor carries a canonical metric. Its Cauchy completion is a final coalgebra for the metric functor and, after forgetting the metric, also for the underlying set functor. We compare this construction with Iterated Function Systems. When the orbit algebra is injective and its comparison with the initial algebra is bi-Lipschitz, the attractor is isomorphic to the completed final coalgebra. This gives a categorical analogue of Hutchinson's theorem for overlaps captured by finitely many distinguished points. Examples include the unit interval, Cantor spaces, the Sierpiński gasket and $p$-adic integers.