# Simplifying the definition of a geometric context using sieves?

On Pages 1-3 of Cours 2 of Toën'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 $P$) 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."

• I agree that someone should get to the category-theoretic bottom of all this (whether or not sieves are the best formal device to use). – JBorger Mar 15 '10 at 5:59
• I just realized that having the axioms in French may be annoying to some of you, so I'll try to come back a little later and reproduce them in English here, but I'm very busy at the moment. – Harry Gindi Mar 15 '10 at 6:10
• Math French is not real French; anyone who intends to go into this kind of algebraic geometry should sit down and learn this kind of French. It takes only a little bit of practice (e.g., comparing English and French versions of a book by Serre, and learning perhaps 30 more words). Reading even a menu in French is harder than reading this kind of French. – BCnrd Mar 15 '10 at 7:49
• I agree. I was just trying to head off any complaints. – Harry Gindi Mar 15 '10 at 8:01

• They're completely trivial to verify for the so-called "algebraic context", which is the affine étale site, where the $P$ is the class of smooth morphisms. They are similarly trivial for topological manifolds, which is the context he discusses in cours 1 and models the general case on (it is not hard to extend this to the case of $C^r$ manifolds or complex manifolds). As I said in the question, it could be that my question is just motivated by my prejudices, but it seems like these very general notions should have simple and elegant definitions. – Harry Gindi Mar 15 '10 at 7:54