Separable and Galois extensions in tensor triangulated categories

I will consider separable and Galois extensions of commutative monoids in tensor triangulated categories, and show how they pop up in various settings.  In stable homotopy theory, separable extensions of commutative S-algebras have been studied extensively by Rognes.
In modular representation theory, restriction to a subgroup can be thought of as extension along a separable monoid in the (stable or derived) module category. In algebraic geometry, separable monoids correspond to étale extensions of schemes, alowing us to define a generalized- étale site for any tensor triangulated category.