Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

Abstract

This paper is concerned with final coalgebra representations of fractal sets. The background to our work includes Freyd’s Theorem: the unit interval is a final coalgebra of a certain endofunctor on the category of bipointed sets. Leinster’s far-ranging generalization of Freyd’s Theorem is also a central part of the discussion, but we do not directly build on his results. Our contributions are in two different directions. First, we demonstrate the connection of final coalgebras and initial algebras; this is an alternative development to one of his central contributions, working with resolutions.Second, we are interested in the metric space aspects of fractal sets. We work mainly with two examples: the unit interval [0,1] and the Sierpiński gasket \(\mathbb{S}\) as a subset of ℝ2.