Timeline for Groups which are only defined up to conjugation
Current License: CC BY-SA 3.0
9 events
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| Mar 1, 2014 at 18:15 | comment | added | Tim Porter | @Qiaochu Yuan: I agree with you in the main, but would mention the use of van Kampen theorem type arguments, at the covering space level, (so Artin gluing I suppose?). I feel that the best way forward theoretically would be to develop an $\infty$-category approach (e.g. look for a natural quasi-category in the context)and then to examine the indeterminacy and the structure of the fillers in the simplicial classes. | |
| Mar 1, 2014 at 17:21 | comment | added | Qiaochu Yuan | @Tim: I mentioned this above but I think the problem with replacing points with fiber functors is that it can still be hard to explicitly write down or work with fiber functors. I think working directly with the category of coverings is the most constructive option. | |
| Mar 1, 2014 at 10:17 | comment | added | Tim Porter | Another idea, to avoid the difficulty that Scott mentions, would be to use more of a Grothendieck SGA1 / Galois theoretic approach to defining the fundamental groupoid (this replaces points by 'fibre functors', so geometric points in a sheaf theoretic setting). The fundamental groupoid then is a clissfying object classifying certain families of coverings of the space. As a universal object it is only defined up to isomorphism, but the property determines it more fully than choosing a base point. This seems nearer to Noah's orginal problem. | |
| Mar 1, 2014 at 8:42 | comment | added | S. Carnahan♦ | Regarding the paragraph on fundamental groups: I think it would be more accurate to say that you need a distinguished basepoint to even define a fundamental group. Otherwise, you have $\pi_0$ on the free loop space, which is just the sort of floppy object that is under discussion. | |
| Mar 1, 2014 at 7:03 | comment | added | Tim Porter | I was also going to suggest localic groupoids, but would then point out that just as you can encode the conjugations of the group in the groupoid, so you can also encode the homotopies in the model so perhaps Noah needs some sort of infinity groupoid model of the setting from the start then collasing back down to something more usual once it is clear what part of it is needed at any given stage. | |
| Mar 1, 2014 at 1:35 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Mar 1, 2014 at 0:08 | comment | added | Zhen Lin | If I'm not mistaken, the opposite of the category of finite extensions of $\mathbb{Q}$, equipped with the "atomic" topology, is a site for the topos of "continuous" $\mathrm{Gal}(\overline{\mathbb{Q}} | \mathbb{Q})$-sets. (One identifies the representable sheaves with the sets with exactly one orbit.) In particular one can define Galois representations to be internal $k$-vector spaces in this topos. It should also be possible to construct a localic groupoid incarnation of $\mathrm{Gal}(\overline{\mathbb{Q}} | \mathbb{Q})$ without choice, using the Joyal–Tierney theory. | |
| Feb 28, 2014 at 23:07 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Feb 28, 2014 at 23:02 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |