Timeline for $2$-categorical structure in Grothendieck's Galois Theory
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 24, 2012 at 20:10 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 235 characters in body
|
| Jun 14, 2012 at 10:15 | comment | added | Martin Brandenburg | Thanks. I've also added a little bit of notation to my question. | |
| Jun 14, 2012 at 7:52 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 179 characters in body
|
| Jun 14, 2012 at 3:37 | comment | added | David Roberts♦ | I've expanded my answer. And I agree that you don't need the groupoid/topos case for this question. | |
| Jun 13, 2012 at 10:40 | comment | added | Martin Brandenburg | I would like to understand groups first before diving into topos theory. Sometimes generalizations are necessary, of course, but I don't see why my question should need this in order to make sense or have a meaningful answer. | |
| Jun 13, 2012 at 5:10 | answer | added | David Roberts♦ | timeline score: 6 | |
| Jun 13, 2012 at 4:46 | comment | added | David Roberts♦ | I would rather think about some sort of groupoid version of this, which is what I gather Grothendieck preferred. Building on Ben Steinberg's comment-answer, you could consider representations of localic groupoids rather than localic groups. But at some point is starts seeming tautologous (though not entirely trivial), especially if you take the view in Johnstone's book Topos Theory where a Galois category is a certain sort of pretopos, and one finds a pro(finite)group so it is a topos of continuous actions. | |
| Jun 13, 2012 at 3:42 | answer | added | Benjamin Steinberg | timeline score: 4 | |
| Jun 12, 2012 at 11:31 | comment | added | Angelo | It seems to me that the 2-category of Galois categories is in fact equivalent to a 1-category. | |
| Jun 12, 2012 at 9:09 | comment | added | Martin Brandenburg | @Sergio: Are you claiming that we have an $2$-adjunction when $\mathsf{TopGrp}^{\mathrm{op}}$ is regarded as a $2$-category with only identity $2$-morphisms? This is also what I was suspecting ... | |
| Jun 12, 2012 at 9:07 | comment | added | Martin Brandenburg | @Scott: Thanks for the link. John Baez' weeks always contain some interesting insights. In this case I already knew how to make $\mathsf{Grp}$ to a $2$-category. But somehow I don't think that this fits here (otherwise I would not have asked). Maybe I'm wrong. | |
| Jun 12, 2012 at 8:38 | comment | added | Buschi Sergio | A 2-cell between $P, Q$ give the the identity when composed by $F'$, then by Godement law on 2-cell composition induce the identity maps on the automorphism groups. | |
| Jun 12, 2012 at 8:20 | comment | added | Buschi Sergio | Only a idea: let $I$ the category $0\to 1$ and $i: I\to Set$ the natural inclusion. Let $A: \mathcal{A}\to Set,\ B: \mathcal{A}\to Set$ and let $F_0, F_1: (A, \mathcal{A})\to (B, \mathcal{B})$, a 2-cell betwenn $F_0$ and $F_1$ is identified by a functor $\phi: (i\times A, I\times \mathcal{A})\to ( B, \mathcal{B})$ in $Cat\downarrow Set$. Then "traslate" this by the adjunction on $TopGrp^{op}$ and see if it can work .. | |
| Jun 12, 2012 at 8:01 | comment | added | S. Carnahan♦ | Have you looked at the 2-category described in math.ucr.edu/home/baez/twf_ascii/week176 ? | |
| Jun 12, 2012 at 5:41 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |