Separability in homotopical algebra

Abstract : Separable algebras are a generalization of étale algebras that can be defined in more general homotopical contexts, and have been studied in tensor-triangular geometry, partly due to their nice properties in this context. In this talk, I will attempt to explain some of these nice properties in tensor-triangulated categories, by showing that they come from surprising features of separable algebras in stable oo-categories, in particular showing that all separable up-to-homotopy algebras lift (almost) uniquely to homotopy coherent algebras. If time permits, I will also mention how the basics of the classical theory of separable algebras extend to homotopical algebra.