The Sierpinski Carpet as a Final Coalgebra Obtained by Completing an Initial Algebra

Abstract: The background for this work includes Freyd's Theorem, in which the unit interval is viewed as a final coalgebra of a certain endofunctor in the category of bipointed sets. Leinster generalized this to a broad class of self-similar spaces in categories of sets, also characterizing them as topological spaces. Bhattacharya, Moss, Ratnayake, and Rose went in a different direction, working in categories of metric spaces, obtaining the unit interval and the Sierpinski Gasket as a final colagebras in the categories of bipointed and tripointed metric spaces respectively. To achieve this they used a Cauchy completion of an initial algebra to obtain the required final coalgebra. In their examples, the iterations of the fractals can be viewed as gluing together a finite number of scaled copies of some set at some finite set of points (e.g. corners of triangles). Here we will expand these ideas to apply to a broader class of fractals, in which copies of some set are glued along segments (e.g. sides of a square). We use the method of completing an initial algebra to obtain the Sierpinski Carpet as a final coalgebra in a category of metric spaces, and note the required adaptations to this approach, most notably that we no longer get the initial algebra as the colimit of a countable sequence of metric spaces. We will explore some ways in which these results may be further generalized to a broader class of fractals. Joint work with Larry Moss.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic