On Pages 1-3 of Cours 2 of Toën's [Master Course on Stacks](http://www.math.univ-toulouse.fr/~toen/cours2.pdf), 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 $P$) 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."



Simplifying the definition of a geometric context using sieves?
First of all, the "extensive list of axioms" is largely a list of *definitions* (of rather natural terminology). Anyway, he seems to be just getting at the issue of the use of fiber squares in arguments involving properties of morphisms, such as come up in many of the basic constructions in algebraic geometry (Hilbert schemes, quotients by group actions, descent theory, etc.). In contrast, one of the points of the sieve language is to carry out Grothendieck topological stuff without assuming the existence of fiber products. So asking for a sieve formulation may be contrary to the spirit of what he is trying to do. The real test is to see how he *uses* this general nonsense before deciding if it is unsuitable for the task to which he intends to apply it. Have you checked out any such uses? 


When I think of all of the *examples* which I care about, his axioms are completely trivial to verify and so the framework seems to just be setting up a way to prove all of the "general nonsense" in one go to later apply it to the usual interesting examples. It only seems long in the same way that the definition of an algebraic group would seem long if one wrote out the definition of every ingredient that goes into it and didn't already know what an algebraic variety is. Is there any interesting *example* for which his actual axioms seem non-trivial to verify? If so, it should be noted in the question. If not, why is the question being asked? (I assume you have at least checked his axioms for etale and fppf morphisms of schemes to see how easily verified and natural they are for "real" examples.) 


