Tannaka duality over ring spectra

Abstract

We prove a Tannaka duality theorem for $(\infty,1)$-categories. This is a duality between certain derived group stacks, or more generally certain derived gerbes, and symmetric monoidal $(\infty,1)$-categories endowed with particular structure. This duality theorem is defined over commutative ring spectra and subsumes the classical statement. We show how the classical theory, and its extension over arbitrary rings, arises as a special case of our more general theory. The application to perfect complexes is explored.