Question
How does one define the complete local ring of an algebraic stack at a (geometric) point?
Including what the right definition might be, this is all I'm asking.
I can do this for schemes and Deligne--Mumford stacks using the pro-representing rings of deformation functors (i.e. functor of maps from artin rings). However, I don't know if these deformation functors are pro-representable if our algebraic stack admits no etale cover.
I present the solution for schemes and DM stacks for those interested.
Solution for schemes
Let $S$ be a locally noetherian scheme and pick a point $s = \operatorname{Spec} k \in S$. Fix an inclusion $k \to \bar k$ and take the corresponding geometric point $\bar s = \operatorname{Spec} \bar k \to S$. Let $\mathfrak{o}_k$ and $\mathfrak{o}_{\bar k}$ be the Cohen rings corresponding to $k$ and $\bar k$ (i.e. the coefficient rings of Cohen structure theorem). Note that the inclusion gives us a flat extension of local rings $\mathfrak{o}_k \to \mathfrak{o}_{\bar k}$.
Now we may define $\hat{\mathcal{O}}_{S,\bar s} := \hat{\mathcal{O}}_{S,s}\otimes_{\mathfrak{o}_k} \mathfrak{o}_{\bar k}$. This is a complete local ring with residue field $\bar k$. But this is cheating and holds no moduli content which we can exploit.
Instead the following observation suggests the general definition. Let $\operatorname{Art}_{\mathfrak{o}_{\bar k}}$ denote the category of local artinian $\mathfrak{o}_{\bar k}$-algebras of finite type over $\bar k$ (hence with residue field $\bar k$). Consider the functor $F: \operatorname{Art}_{\mathfrak{o}_{\bar k}} \to (\text{Sets})$ defined by $A \mapsto \hom_{\bar s}(\operatorname{Spec} A, S)$, where the subscript $\bar s$ indicates that the morphisms to $S$ restrict to $\bar s \to S$ over the residue field of $A$.
It is easy to see that $\hat{\mathcal{O}}_{S,\bar s}$ pro-represents $F$. This should be our definition in general.
Solution for DM stacks
Let $X \to S$ be a an algebraic stack defined over a base scheme $S$. Suppose $\bar s = \operatorname{Spec} \bar k \to S$ is a geometric point and $\bar x = \operatorname{Spec} \bar k \to X$ is a geometric point of $X$ lying over $\bar s$. Pick an etale cover $U \to X$ and a geometric point $\bar u \to U $ lying over $\bar x$.
This time consider the functor $F:\operatorname{Art}_{\hat{\mathcal{O}}_{S,\bar s}} \to (\text{Sets}) : A \mapsto \hom_{\bar x}(\operatorname{Spec} A,X)$. The geometric formal neighbourhood of a scheme is defined above, the notation $\hom_{\bar x}$ should really mean that we consider morphisms from $\operatorname{Spec} A$ which is 2-isomorphic to $\bar x \to X$ \emph{and} a 2-isomorphism between the two things.
Because $U \to X$ is etale, $F$ is isomorphic to the funtor $A \mapsto \hom_{\bar u}(\operatorname{Spec} A,U)$ which we studied above. So $F$ is pro-representable with representing ring $\hat{\mathcal{O}}_{U,\bar u}$.
In particular, the ring pro-representing $F$ can be defined without reference to an atlas. We can simply denote this ring by $\hat{\mathcal{O}}_{X,\bar x}$ and call it the complete local ring of the geometric point $\bar x$.
