The Sierpinski Carpet as a Final Coalgebra

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 metric spaces 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 exists for our functor, and that the final coalgebra is bilipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as ω-colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson [7].