There are a number of notions of "cover" for a scheme: etale, faithfully flat, fpqc, fppf, Zariski, Nisnevich, etc. Most of these have a nice property, which is that a cover of that type satisfies effective descent for modules or quasicoherent sheaves. For me it will be easiest to state this just for rings (hence for affine objects): A morphism $f:R\to S$ of commutative rings satisfies effective descent for modules if the category of $R$-modules can be recovered from the category of $S$-modules with descent data. There are many interpretations of descent data, but it should basically be thought of as gluing data for $Spec(S)\times_{Spec(R)} Spec(S)$.
Grothendieck showed that faithfully flat morphisms always satisfy this condition, and Joyal and Tierney showed that a more general class of morphisms, called pure morphisms, completely classifies the morphisms which satisfy this condition (i.e. a morphism satisfies effective descent for modules if and only if it is "pure"). Does anyone know what the "pure" site looks like? Is it subcanonical? How come nobody ever uses it for anything?
More generally, suppose I have some other stack (i.e. not the stack of modules, but something else). I can associate to it a topology where the covers are exactly the morphisms along which this stack descends. What does this topology look like? Has anyone studied it?
