On Pages 1-3 of Cours 2 of Toen'sToën's Master Course on Stacks, he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over and above the definition of a Grothendieck topology, which isn't even included). Maybe it's just my prejudices talking, but it seems like there should be a way to simplify this definition using sieves or some other kind of functorial machinery (think about the definition of a sheaf in terms of sieves compared to the definition using the gluing axiom). The reason I ask this is that the important properties of class the geometric structure morphisms (what Toën calls P$P$) allow us to define closed and open subsheaves, representable covers, atlases, etc, which all seem like things that sieves were meant to do.
Question: Can we simplify the statements of those axioms using sieves or some other kind of functorial machinery? If not, why not?
Edit: I just remembered that "geometric morphism" already means something else, so I've replaced it. "The name 'geometric structure morphism' is a word that I coined myself, spending a week thinking of nothing else."