On the Unipotent $p$-adic Simpson Correspondence

The goal of this paper is to show a (derived) $p$-adic Simpson correspondence for (locally) unipotent coefficients on smooth rigid-analytic varieties. Our results depend on a deformation to ${\mathbf{B}}_{\mathtt{d}\mathtt{r}}^{+}\mathrm{/}{\xi }^2$, and not on a choice of exponential (as required for more general coefficients). Our methods are inherently higher categorical, hinging on the theory of modules over ${\mathbf{E}}_{\mathrm{\infty }}$-algebras.