The $p$-adic integers as final coalgebra

TL;DR

This paper models the classical $p$-adic integers as a final coalgebra in a categorical setting, providing a new perspective on their algebraic and metric structure.

Contribution

It introduces a novel coalgebraic representation of $p$-adic integers, connecting their algebraic operations with categorical coalgebra concepts.

Findings

$p$-adic integers are expressed as a final coalgebra.

Addition and multiplication are realized as coalgebra maps.

Provides a categorical framework for $p$-adic number operations.

Abstract

We express the classical $p$-adic integers $\hat{\mathbb{Z}_p}$, as a metric space, as a final colagebra to a certain endofunctor. We realize the addition and the multiplication on $\hat{\mathbb{Z}_p}$ as the coalgebra maps from $\hat{\mathbb{Z}_p} \times \hat{\mathbb{Z}_p}$.