Zp as a final coalgebra obtained by Cauchy completing the initial algebra

Abstract

The metric space of p -adic integers with the p -adic metric, Zp , is presented as a final coalgebra obtained as the Cauchy completion of the initial algebra of an endofunctor on the category of one-pointed one-bounded metric spaces with short maps. This fact, that the final coalgebra is the Cauchy completion of the initial algebra, is used to show that Zp is also the final coalgebra of this endofunctor in the continuous setting. Some final coalgebras on pointed metric spaces with short maps are known to be the Cauchy completion of the intial algebra. In a separate study, Zp has been observed as the final coalgebra of certain endofunctors on ultra metric spaces. The results of this paper unify these observations and gives a coalgebraic characterisation of the self similarity of Zp, while relaxing the ultra metric condition to one-bounded metrics. Obtaining the final coalgebra as the Cauchy completion of the initial algebra is in close analogy with classical results in iterated function systems. Another question that has been asked in the literature is whether such results hold when the maps are chosen to be Lipchitz. We give evidence as to why selecting continuous maps as the morphisms may be the right choice for such results to hold.