Most of my exposure to modules has been over the various forms of category algebras, such as group algebras, path algebras, and incidence algebras. As such, I don't have a lot of exposure to modules over arbitrary rings. Right now, I am trying to understand modules over rings of formal power series $k[\![X]\!]$, and I'm wondering if there's a way to understand them as representations of a particular object. I know for $k[X]$ we may envision $k[X]\!\operatorname{-Mod}$ as representations of $\mathbb{N}^{|X|}$, but can we make any such claim for $k[\![X]\!]$?
More precisely is there a category, $C$ such that $k[C] \cong k[\![X]\!]$? If so, does $C$ reasonably correspond to some more common combinatorial or algebraic object like a quiver or a monoid? Otherwise, why not? Is there something about $k[\![X]\!]$ that prevents this from happening?
Note, I'm interested in the case of $X$ potentially being infinite, however if it is true only of finite $X$ I would be interested in hearing why.
Edit: relatively quickly after posting this, I realized $k[\![X]\!]$ has no non-trival idempotents, so $C$ must have a single object, so we can ignore the category theory and simply ask is their a monoid $M$ such the $k[\![X]\!] \cong k[M]$?
No, they're never monoids algebras.
In the paper Monoid rings which are valuation rings, J. Ohm, P. Vicxnair, they mention such a theorem as a "well known result".
Theorem (folklore). If $k[M]$ is a valuation ring, then $k$ is a characteristic $p$ field, and $M$ is either $\Bbb Z/p^r$ or $\Bbb Z[1/p]/\Bbb Z$.
Complete proof of this statement is a bit involved, but fairly elementary, and can be found in this paper.
Formal power series are not monoid rings, because they are uncountable-dimensional valuation rings for any base field and any number of variables, while all monoids resulting in valuation rings are countable.