On the Initial Algebra and Final Co-algebra of some Endofunctors on Categories of Pointed Metric Spaces

TL;DR

This paper investigates the initial algebra and final co-algebra of specific endofunctors on categories of pointed metric spaces, revealing their structures and limitations under various morphism types, including continuous, short, and Lipschitz maps.

Contribution

It characterizes the final co-algebra for these endofunctors in different metric space categories and shows the Sierpinski gasket is not the final co-algebra under Lipschitz maps.

Findings

Final co-algebra coincides with the set-theoretic one when using continuous or short maps.

The co-algebra can be constructed via a Cauchy completion process.

The Sierpinski gasket is not the final co-algebra with Lipschitz maps.

Abstract

We consider two endofunctors of the form $~F:X\longrightarrow M\otimes X~$, where $~M~$ is a non degenerate module, related to the unit interval and the Sierpinski gasket, and their final co-algebras. The functors are defined on the categories of bi-pointed and tri-pointed metric spaces, with continuous maps, short maps or Lipschitz maps as the choice of morphisms. First we demonstrate that the final co-algebra for these endofunctors on the respective category of pointed metric spaces with the choice of continuous maps is the final co-algebra of that with short maps and after forgetting the metric structure is of that in the set setting. We use the fact that the final co-algebra can be obtained by a Cauchy completion process, to construct the mediating morphism from a co-algebra by means of the limit of a sequence obtained by iterating the co-algebra. We also show that the Sierpinski gasket $~(\mathbb{S},\sigma)~$ is not the final co-algebra for these endofunctors when the morphism are restricted to being Lipschitz maps.