Appears in collection : Arithmetic Geometry – A Conference in Honor of Hélène Esnault on the Occasion of Her 70th Birthday

Prismatic cohomology is a unifying p-adic cohomology of $p$-adic formal schemes. Motivated by questions on locally analytic representations of $p$-adic groups and the $p$-adic Simpson correspondence, an extension of prismatic cohomology to rigid-analytic spaces (over $Q_p$ or over $F_p((t)))$ has been sought. We will explain what form this should take, and our progress on realizing this picture. This includes a degeneration from the analytic Hodge-Tate stack underlying the $p$-adic Simpson correspondence to a similar (analytic) stack related to the Ogus-Vologodsky correspondence in characteristic $p$. This is joint work in progress with Johannes Anschütz, Arthur-César le Bras and Juan Esteban Rodriguez Camargo.