Positselski duality in $\infty$-categories

We introduce the notion of a contramodule over a cocommutative coalgebra in a presentably symmetric monoidal $\infty$-category $\mathcal{C}$, and prove a symmetric monoidal $\infty$-categorical version of Positselski's comodule-contramodule correspondence when the coalgebra is coidempotent. This gives a new perspective on, and a new proof of local duality -- in the sense of Hovey--Palmieri--Strickland and Dwyer--Greenlees -- whenever $\mathcal{C}$ is stable and compactly generated. We further consider an analog of Positselski's definition of contramodules over topological rings in the $\infty$-categorical setting, and show that the two perspectives on contramodules are equivalent. As examples we describe the categories of $K(n)$-local spectra, $T(n)$-local spectra and the derived complete category of a ring $R$, as categories of contramodules.