Download PDFOpen PDF in browserFractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra5 pages•Published: July 28, 2014AbstractThis 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 certain functor on the category of bipointed sets. Leinster 2011 offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called "selfsimilar" spaces using (a) bimodules (also called profunctors or distributors), (b) an examination of nondegeneracy 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 omegasequence of finite metric spaces. This generalizes Hutchinson's 1981 characterization of fractal attractors 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 \emph{metric space} characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010, and our interest in quotient metrics comes from their paper. So in terms of (a)(c) above, our work develops (a) and (b) in metric settings while dropping (c). Keyphrases: bimodule, final coalgebra, fractal set, initial algebra, metric space In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 158162
