The Sierpinski Carpet as a Final Coalgebra

TL;DR

This paper explores the Sierpinski carpet as a final coalgebra, establishing a categorical framework that connects fractal geometry with coalgebraic structures, and demonstrating their equivalence.

Contribution

It constructs a category of square sets and an endofunctor, proving the existence of initial algebra and final coalgebra, with the latter being bi-Lipschitz equivalent to the Sierpinski carpet.

Findings

Final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet.

Constructs a category of square sets and an associated endofunctor.

Establishes the existence of initial algebra and final coalgebra for the functor.

Abstract

We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit square. We construct a category of square sets and an endofunctor on it which corresponds to the operation of gluing copies of a square set along segments. We show that the initial algebra and final coalgebra exist for our functor, and that the final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson.