Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

5 pagesPublished: July 28, 2014

Abstract

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 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 "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 omega-sequence 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 158--162