Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra Lawrence S. Moss * , Jayampathy Ratnayake, and Robert Rose Mathematics Department, Indiana University, Bloomington, IN 47405 USA Abstract This paper is a contribution to the presentation of fractal sets in terms of final coal- gebras. The first result on this topic was Freyd’s Theorem: the unit interval [0, 1] is the final coalgebra of a functor X → 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 dis- tributors), (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). 1 Introduction As our abstract above indicates, this paper is largely a kind of “marginal note” to Leinster’s paper on self-similar spaces obtained as final coalgebras of a certain sort. Our 4-page TACL abstract cannot even hope to present all of Leinster’s definitions and central results, and so we must refer the reader to [L] for all of the background. In fact, we are going to take the unusual step of not even presenting all of our own definitions and results, since a reader unfamiliar would not be able to follow a brief presentation, and a reader who has studied [L] and/or [HJN] would likely see our general formulations. In addition to the “marginal note” on using completions of initial algebras, we are interested in metric-space versions of non-degeneracy conditions and recognition theorems from [L]. Our contributions here involve borrowings from metric geometry (see [BH]). And so our remarks in this abstract are mainly hints as well. Now that we have said mentioned that this abstract is not a decent presentation of our results, let alone of Leinster [L], we should say what we are presenting. We are going to give a a concrete example of how our constructions work, aimed mainly at the uninitiated reader. At times we shall of course mention connections to other papers, especially Leinster [L]. * This work was partially supported by a grant from the Simons Foundation (#245591 to Lawrence Moss). 158 N. Galatos, A. Kurz, C. Tsinakis (eds.), TACL 2013 (EPiC Series, vol. 25), pp. 158–162