Timeline for Fundamental groups of topoi
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Feb 5, 2011 at 11:01 | comment | added | David Roberts♦ | AH, hmm, you're right. Maybe it's only Galois topoi for which this is true (I'm guessing a name based on a half-remembered concept, so bear with me in my folly) | |
| Feb 5, 2011 at 9:12 | comment | added | Mike Shulman | I don't think that just because a topos is the topos of sheaves on a localic groupoid implies that that localic groupoid is the fundamental groupoid of the topos. The localic groupoids that occur as fundamental groupoids of topoi are all prodiscrete, whereas any localic groupoid has a topos of sheaves. | |
| May 7, 2010 at 16:40 | comment | added | Peter Arndt | ... in some T/X_i after pullback. Now you say that Z in T locally has a property if for some covering in the above sense all the pullbacks of Z along the geometric morphisms of the covering have this property. (I think actually it is enough to consider a single slice topos which is a covering, since in the above situation you can always form the coproduct of the X_i) | |
| May 7, 2010 at 16:35 | comment | added | Peter Arndt | The notion of something being locally true for an object Y in a topos T is: You look at a collection of objects X_i and the slice categories T/X_i (these are topoi again, corresponding to open subsets). You get a collection of geometric morphisms T/X_i-->T (--> given by composing Z->X_i with X_i->1 [1 the terminal object of the topos which always exists], <-- given by associating with Z' the pullback along X_i-->1, i.e. product with X_i+projection). You should see this as a covering if the familiy <-- is jointly conservative, i.e. if two arrows are different in T then they will be so in ... | |
| May 6, 2010 at 6:51 | comment | added | Lars | Unfortunately, if I remember correctly, even the cyrstalline site of a non smooth scheme, does not need to have a final object. And it would be interesting to know the fundamental group of the crystalline topos. | |
| May 4, 2010 at 23:41 | comment | added | David Roberts♦ | An object $Y$ in a topos $E$ is called locally constant if there is a cover $U$ of the terminal object and an isomorphism $Y \times U \simeq U \times p^*F$ for some set $F$ and where $p:E \to Set$ is the canonical map. I think for the sort of site you seem to interested in, there is a terminal object (small site of a scheme/space), so that's not too much of a complicating factor, I suppose. | |
| May 4, 2010 at 16:26 | comment | added | Lars | Hi, thanks for the answer. Unfortunately I'm not very fluent in Topoi-language, I will look up what a locale is though. I was aware of the problem that sites don't need to have final objects, that's why I was sketchy :) Maybe one can only talk about $Y$-local properties then, for any $Y$ in the site. Or maybe one embeds the site via yoneda into the topos and uses the a covering of the final object in the topos. | |
| May 4, 2010 at 14:31 | history | answered | David Roberts♦ | CC BY-SA 2.5 |