Sierpinski Gasket as a Final Coalgebra Obtained by Cauchy Completing the Initial Algebra

Abstract

This paper presents the Sierpinski Gasket ($\mathbb{S}$) as a final coalgebra obtained by Cauchy completing the initial algebra for an endofunctor on the category of tri-pointed one bounded metric spaces with continuous maps. It has been previously observed that $\mathbb{S}$ is bi-Lipschitz equivalent to the coalgebra obtained by completing the initial algebra, where the latter was observed to be final when morphisms are restricted to short maps. This raised the question "Is $\mathbb{S}$ the final coalgebra in the Lipschitz setting?". The results of this paper show that the natural setup is to consider all continuous functions. The description of the final coalgebra as the Cauchy completion of the initial algebra has been explicitly used to determine the mediating morphism from a given coalgebra to the the final coalgebra. This has been used to show that if the structure map of a coalgebra is continuous, then so is the mediating morphism. The description of $\mathbb{S}$ given here not only generalizes previous observations, but also unifies classical descriptions of $\mathbb{S}$. We also show, by means of an example, that $\mathbb{S}$ is not the final coalgebra if we consider only Lipschitz maps.