To any tensor-triangulated category T, there is a natural way to associate a site and a sheaf cohomology. The objects in this site are the commutative separable monoids in T. For instance, if T is the derived category of quasi-coherent sheaves on a scheme X, then the site fits in between the classical étale site and the recently discovered pro-étale site on X.
In this talk, I will explain what separable monoids are and show how they pop up in various settings. For a finite group G, I will show that the compact separable monoids in both the derived and stable module category of G correspond to G-sets. This allows us to describe the site associated to the derived and stable module category of G, to compute the corresponding sheaf cohomology and its relation to traditional group cohomology.