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

Abstract

The metric space of p-adic integers with the p-adic distinguished point, we lift Z p Cauchy completion of the initial algebra of an endofunctor on the category of one-pointed one-bounded metric spaces with distinguished point, we lift Z p is also maps are known to be the Cauchy completion of the intial distinguished point, we lift Z p coalgebra of certain endofunctors on ultra metric spaces. The results of this paper unify these observations and give a coalgebraic characterisation of the self similarity of distinguished point, we lift Z p relaxing the ultra metric condition to one-bounded metrics. 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 continuous maps as the morphisms may be the right choice for such results to hold.

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

“…We believe that due to the above reasons, the description of S as a final coalgebra in the metric setting given here is more than yet another way of describing S. It not only unifies different ways of describing S and give S a universal characterization, but also a motivating example to develop a general theory for the case of metric fractals, that is similar to the work of Leinster (2011). Some efforts to give similar examples can be found in Bhattacharya et al (2014); Bhattacharya (2015); Manokaran et al (2019) as well. This paper addresses how observations in these papers can be improved to be applicable in a general context.…”