Timeline for Simplifying the definition of a geometric context using sieves?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2010 at 4:11 | vote | accept | Harry Gindi | ||
| Mar 15, 2010 at 15:37 | comment | added | BCnrd | @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"? | |
| Mar 15, 2010 at 8:32 | comment | added | Harry Gindi | 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. | |
| Mar 15, 2010 at 8:17 | comment | added | BCnrd | @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. | |
| Mar 15, 2010 at 7:54 | comment | added | Harry Gindi | 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 7:38 | history | answered | BCnrd | CC BY-SA 2.5 |