Deformation Theory of $\mathbb{E}_n$-Monoidal Categories

In this paper, we prove that the naive deformation problem of an $\mathbb{E}_n$-monoidal stable $k$-linear $\infty$-category $\mathcal{C}$ is a $2$-proximate formal $\mathbb{E}_{n+2}$-moduli problem, whose corresponding formal moduli problem is controlled by the non-unital $\mathbb{E}_{n+2}$-algebra $\mathrm{fib}\big(\mathrm{End}_{\mathcal{Z}_{\mathbb{E}_n}(\mathcal{C})}(1)\rightarrow \mathrm{End}_{\mathcal{C}}(1)\big)$, where $\mathcal{Z}_{\mathbb{E}_n}(\mathcal{C})$ is the $\mathbb{E}_n$-center of $\mathcal{C}$. If $\mathcal{C}$ is rigid monoidal and tamely compactly generated by unobstructible objects, then this naive deformation problem is equivalent to the formal moduli problem. We also prove a uniqueness theorem for formal deformations of certain formal moduli problems, which can be applied to the $\mathbb{E}_1$ and $\mathbb{E}_2$-monoidal deformation problems of $\mathbf{Rep}(G)$ for a reductive algebraic group $G$ with a simple Lie algebra $\mathfrak{g}=T_e G$. Finally, we show factorization homology is compatible with deformations.