Timeline for Do strict pro-sets embed in locales?
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| Jun 28, 2011 at 2:12 | comment | added | Benjamin Steinberg | Moerdijk gives examples of the same sort showing that a profinite groupoid is not the same thing as a groupoid in Stone Spaces in the AMS memoir Proper maps of toppers. He basically shows if R s a closed equivalence relation on a Stone space X which is not open, then the associated groupoid is not profinite. He argues by looking at the classifying topos, but one can see it directly. | |
| Feb 3, 2011 at 6:26 | history | edited | Dmitri Pavlov |
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| Jan 16, 2011 at 5:54 | answer | added | Mike Shulman | timeline score: 3 | |
| Dec 15, 2010 at 19:14 | comment | added | Mike Shulman | @Theo: You seem to be getting into very different territory then! Anyway, would you care to add the example to something like ncatlab.org/nlab/show/profinite+groupoid ? | |
| Dec 15, 2010 at 7:55 | comment | added | Theo Johnson-Freyd | And when I say "hom of topological groupoids" I don't mean a topological functor, but a left-principal bibundle, because when I say "groupoid" I really mean "stack that it represents". | |
| Dec 15, 2010 at 7:48 | comment | added | Theo Johnson-Freyd | Er, I should also emphasize that everywhere I said "functor" I meant "2-functor", of course. I'm claiming that the two groupoids are equivalent (not isomorphic) as pro(finite groupoids), but not equivalent as topological groupoids. | |
| Dec 15, 2010 at 7:46 | comment | added | Theo Johnson-Freyd | @Mike: The only place I know the example to be written down is in Alex's emails to me. The example came up because we've been thinking about constructions in the neighborhood of "etale pi_1", which is already a profinite construction (algebraic geometry can't "see" truly infinite things). The reason these represent the same pro(finite groupoid) is because both are "the" fundamental groupoid of the circle. Oh, and thanks for the Johnstone reference! | |
| Dec 15, 2010 at 4:29 | comment | added | David Roberts♦ | Hmm - at one point I thought it would be worthwhile thinking not about Pro(finite groupoids) but Pro(weakly finite groupoids), where by the latter I mean groupoids with finite $\pi_0$ and $\pi_1$ for all basepoints. Does your example hold up if you consider the prorepresentable functors from weakly finite groupoids to groupoids? | |
| Dec 14, 2010 at 18:00 | comment | added | Mike Shulman | By the way, since StoneTop is equivalent to Pro(FinSet), it must be that Gpd(StoneTop) is equivalent to Gpd(Pro(FinSet)). I guess the point is that neither of these is equivalent to Pro(Gpd(FinSet)), in contrast to how Grp(Pro(FinSet)) is equivalent to Pro(Grp(FinSet)). It may be worth noting that in VI.2 of Stone spaces, Johnstone gives general conditions on a finitary algebraic theory T which ensure that T-Alg(Pro(FinSet)) is equivalent to Pro(T-Alg(FinSet)), along with some T for which this fails. Of course groupoids are not a finitary algebraic theory in this sense. | |
| Dec 14, 2010 at 17:05 | comment | added | Mike Shulman | That's very interesting, thanks! Is the proof that those two Stone-topological groupoids represent the same pro(finite groupoid) written down somewhere? I sure don't see the reason immediately. | |
| Dec 14, 2010 at 0:01 | comment | added | Theo Johnson-Freyd | (continuation) from finite groupoids (thought of as topological groupoids) to groupoids. I.e. they represent the same pro(finite groupoid). I learned this counterexample from Alex Chirvasitu, with whom I'm doing a joint project and we would be much happier if they were the same. (There are some old papers that seem to say the question of whether (pro(finite groupoid)) = ((profinite) groupoid) was open; I don't know if this example is new to Alex.) So the point is: there's something special about the result for groups. ((Oh, and "Chech" in my previous comment is of course "Cech".)) | |
| Dec 13, 2010 at 23:58 | comment | added | Theo Johnson-Freyd | A word of caution. It is not true that the (2-)category of pro(finite groupoids) is equivalent to the (2-)category of groupoids in the category of profinite topologies. An example is the following. Take, for example, the circle $S$. Let $X$ be the Chech compactification of the discrete space under $S$, and $X\to S$ the canonical surjection. Then $X \times_S X \rightrightarrows X$ is a groupoid in (profinite spaces), which is not equivalent to $\hat{\mathbb Z}\rightrightarrows 1$, the profinite completion of the integers. But they represent the same functor (continued) | |
| Dec 13, 2010 at 20:36 | history | asked | Mike Shulman | CC BY-SA 2.5 |