Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A **braided deformation** of $\mathcal{C}$ over a local artin ring $R$ with residue field $k$ is an $R$-linear braided monoidal category $\mathcal{C}'$, whose hom-sets are free $R$-modules, together with an equivalence of braided monoidal categories $\mathcal{C}' \otimes_{R} k \simeq \mathcal{C}$.

If I've understood correctly, there is a deep result of Drinfeld-Cartier that states that braided deformations satisfy a degree of "unobstructedness." More specifically, any deformation over $k[\epsilon]/\epsilon^2$, which is trivial as a deformation of monoidal categories, can be lifted to a formal deformation over $k[[\epsilon]]$. The proof starts by considering the "cocycle" that a braided deformation over the dual numbers yields, and then exponentiates it to get the braiding over $k[[\epsilon]]$. Unfortunately, that will not satisfy the hexagon axiom (because the exponential of matrices is not a homomorphism in general), so one has to modify the associativity constraint via "Drinfeld associators."

I'm curious if this result can be understood from a more homotopy-theoretic standpoint. For example, I'd like to know if there is an analogous picture in higher category theory.

- A possible analog of $\mathcal{C}$ would be a $k$-linear presentable symmetric monoidal, stable $\infty$-category in the sense of Lurie. One benefit of working with these is that there is a good theory of tensor products and base change.
The deformation problem could be the following: over a local artin ring $R$ with residue field $k$, consider all $R$-linear $E_2$-categories $\mathcal{C}'$ (presentable, stable) $\mathcal{C}$ equipped with an equivalence of $E_2$-categories $\mathcal{C}' \otimes_R k \simeq \mathcal{C}$. This defines a deformation problem $R \mapsto \mathrm{Def}(R)$.

Then a natural question would be: is this deformation problem unobstructed in general? Or, at least, is every first-order deformation trivialized in the $E_1$-direction liftable to a formal deformation?

- Are there generalizations of this to $E_n$-monoidal (presentable, stable) $\infty$-categories?