Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For example, the category of monoids is one way to 'model' algebras over field with one element or $\mathbb{F}_1$, due to A. Deitmar building on the ideas of K. Kato. The $\mathbb{F}_1$-algebras or affine $\mathbb{F}_1$-schemes are monoids and the base change functor to $\mathbb{Z}$-algebras sends a monoid $A$ to the monoid algebra $\mathbb{Z}[A]$. There is a notion of 'monoidal space' analogous to ringed space and a notion of $\mathbb{F}_1$-scheme that is a gluing of affine $\mathbb{F}_1$-schemes.
In this category, we can think of $\mathbb{F}_1[t]$ as $\mathbb{N}$, for instance by thinking of it as the free monoid on one generator.