# 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."

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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

## 1 Answer

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.)

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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
@fpqc: my point is that these very general notions already are simple and elegant for many purposes, and that the question should provide a specific example of a context for which the axioms would be interesting to have but as formulated are unpleasant to verify or not applicable (or false). In the absence of an applicable setting, there's nothing to serve as a criterion for a "simplification". We don't make up axioms in a vacuum. You could be wondering "can this all be expressed using sieves", but then just pose that 1-line question; comments about "extensive list..." aren't relevant. – BCnrd Mar 15 '10 at 8:17
I guess that the main point of what I was asking was if we could express these notions more economically. The best example I can think of is effective descent on a cover for a fibered category, where there are compatibility conditions that require a choice of cleavage or distinguished fiber products, but we can express all of those conditions more elegantly (and without making choices) using sieves. – Harry Gindi Mar 15 '10 at 8:32
@fpqc: Toen's setup seems aimed at descent for properties of morphisms (not descent for objects), and for that his notions are very economical. The most interesting examples are properties far from covering morphisms, such as etale descent for fppf, surjectivity, finitely presented with fibers of pure dimension 14, etc. You never address how is Toen using the axioms, yet that is what dictates how to recast them if you wish; otherwise it's making up axioms in a vacuum, which is pointless. What is your "context" for wanting to read about stuff as arid as a "geometric context"? – BCnrd Mar 15 '10 at 15:37