Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras. The first result on this topic was Freyd’s Theorem: the unit interval [0, 1] is the final coalgebra of a functor X 7→ X ⊕ X on the category of bipointed sets. Leinster [L] offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called self-similar spaces using (a) bimodules (also called profunctors or distributors), (b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction of final coalgebras for the types of functors of interest using a notion of resolution. In addition to the characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces. Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions is not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces of interest as the Cauchy completion of an initial algebra, and this initial algebra is the set of points in a colimit of an ω-sequence of finite metric spaces. This generalizes Hutchinson’s characterization of fractal attractors in [H] as closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is “computationally related” to the overall fractal. For example, when applied to Freyd’s construction, our method yields the metric space of dyadic rational numbers in [0, 1]. Our second contribution is not completed at this time, but it is a set of results on metric space characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui [HJN], and our interest in quotient metrics comes from [HJN]. So in terms of (a)–(c) above, our work develops (a) and (b) in metric settings while dropping (c).