On the deformation theory of $\mathbb{E}_\infty$-coalgebras

We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect $\mathbb{F}_p$-algebra $k$, Weil restriction along the augmentation $\mathbb{W}(k)\to k$ induces a fully faithful functor from formally \'etale, connective $\mathbb{E}_{\infty}$-coalgebras in $k$-modules to connective $\mathbb{E}_{\infty}$-coalgebras in $p$-complete modules over the spherical Witt vectors $\mathbb{W}(k)$. Finally, we prove that for any connected space $X$, the $k$-homology $k[X]$ is a formally \'etale $\mathbb{E}_\infty$-coalgebra in $k$-modules. This shows that $\mathbb{W}(k)[X]^{\wedge}_p$ can be recovered as the essentially unique lift of $k[X]$ to a connective coalgebra in $p$-complete $\mathbb{W}(k)$-modules.