The prismatization of $p$-adic formal schemes

In this note, we introduce and study the Cartier–Witt stack ${\mathrm{W}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{t}}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on ${\mathrm{W}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{t}}_X$ in favorable situations.