Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

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 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 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).

Presenting the Sierpinski Gasket in Various Categories of Metric Spaces

“…We next present the functor that is central to the discussion of this paper. Definitions below were considered previously in [Moss et al, 2013, Bhattacharya et al, 2014, and the notation follows [Leinster, 2011].…”

“…The main results in this paper concern the characterizations of the Sierpinski gasket in metric terms found in [Moss et al, 2013, Bhattacharya et al, 2014. A tripointed metric space (X, d) is a tripointed set (X, T, L, R) equipped with a 1-bounded metric d (i.e., d(x, y) ≤ 1 ∀ x, y ∈ X), such that the distance between any pair of distinguished elements is 1.…”

“…Leinster generalized Freyd's work to provide a general theory of self-similarity, aiming at the a deeper understanding of fractal subsets of real or complex spaces. The work in this paper belongs to a set of papers [Hasuo et al, 2010, Moss et al, 2013, Bhattacharya et al, 2014, Noquez and Moss, 2021] that add metric information to Freyd's characterization and go on to consider metric characterizations of other fractals, and also are influenced by Leinster's work. When one considers categories of metric spaces, it is important to settle on the choice of morphisms.…”

“…We study the initial algebra of F and also the final coalgebra of F in all settings. The first two rows of the chart were established in [Moss et al, 2013, Bhattacharya et al, 2014, and the bottom two rows are new here.…”

“…Since we are mainly interested in fractal sets characterized as final coalgebras, the reader might wonder why we also mention initial algebras. The reason is that the results in [Moss et al, 2013, Bhattacharya et al, 2014 construct the final coalgebra as the Cauchy completion of the initial algebra (with the inverse structure). Incidentally, this is an usual occurrence: one can find examples where the initial algebra and final coalgebra are the same object with inverse structures, but what we have in this area seems rare.…”