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

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.