Report on ${\mathbb{E}}_{\mathrm{\infty }}$-descendability

We introduce the notion of ${\mathbb{E}}_{\mathrm{\infty }}$-descendability as well as a derived variant. We prove that several classes of descendable maps of commutative rings are ${\mathbb{E}}_{\mathrm{\infty }}$-descendable. As an application, we prove a variant of Tannaka duality.