Presenting the Sierpinski Gasket in Various Categories of Metric Spaces

TL;DR

This paper explores how the Sierpinski gasket can be represented as a final coalgebra across different categories of metric spaces, revealing the existence and nature of such representations depending on the morphisms used.

Contribution

It introduces a unified framework for presenting the Sierpinski gasket as a final coalgebra in various metric space categories, highlighting differences based on morphism types.

Findings

Final coalgebra exists in the continuous setting as the gasket itself.

In the short setting, the final coalgebra is the completion of the initial algebra, not isomorphic to the gasket.

No final coalgebra exists in the Lipschitz setting.

Abstract

This paper studies presentations of the Sierpinski gasket as a final coalgebra for functors on several categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, a second uses Lipschitz maps, and a third uses continuous maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. We prove that the Sierpinski gasket itself is the final coalgebra of a naturally-occurring functor in the continuous setting. In the short setting, the final coalgebra exists but it is better described as the completion of the initial algebra, and this is not isomorphic to the Sierpinski gasket. In the Lipschitz setting, the final coalgebra does not exist. We determine the initial algebras in all three settings as well.