# 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). Mar 15, 2010 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. Mar 15, 2010 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. Mar 15, 2010 at 7:49
• I agree. I was just trying to head off any complaints. Mar 15, 2010 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. Mar 15, 2010 at 7:54