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

Abstract

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 $\mathrm{fib}({\mathrm{End}}_{{\mathcal{Z}}_{{\mathbb{E}}_n}(\mathcal{C})}(1)\to {\mathrm{End}}_{\mathcal{C}}(1))$, 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.