On the deformation theory of $\mathbb{E}_\infty$-coalgebras
We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras and show that these admit essentially unique, functorial lifts against square zero extensions of $\mathbb{E}_{\infty}$-rings. We use this to construct a spherical Witt vector style functor which exhibits the $\infty$-category of formally \'etale connective $\mathbb{E}_{\infty}$-coalgebras over $\mathbb{F}_{p}$ as a full subcategory of the $\infty$-category of connective $p$-complete $\mathbb{E}_{\infty}$-coalgebras over $\mathbb{S}_{p}^{\wedge}$. Finally, we prove that for a finite space $X$ the $\mathbb{F}_{p}$-homology $C_{\ast}(X;\mathbb{F}_{p})$ is a formally \'etale $\mathbb{F}_{p}$-coalgebra and hence $(\Sigma^{\infty}_{+}X)^{\wedge}_{p}$ can be recovered as its essentially unique lift to the $p$-completed sphere.