Timeline for Is the fundamental group functor a left-adjoint?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2021 at 2:08 | vote | accept | ziggurism | ||
| Oct 17, 2012 at 5:22 | comment | added | HJRW | The OP seems to be under a misapprehension. The hypotheses can be (and usually are) weakened: the inclusion maps are not required to be $\pi_1$-injective. See, for instance, Wikipedia: en.wikipedia.org/wiki/Seifert–van_Kampen_theorem . | |
| Oct 17, 2012 at 0:46 | comment | added | Marc Hoyois | @Davidac: The étale fundamental pro-groupoid functor is also a left adjoint functor from the étale $\infty$-topos to pro-groupoids, and it therefore preserves all homotopy colimits. In particular, if you have any étale cover of a scheme $X$, then the étale fundamental pro-groupoid of $X$ is the 2-colimit of the diagram formed by the pro-groupoids of all the finite "intersections" of the schemes in the cover. | |
| Oct 16, 2012 at 21:43 | comment | added | David Corwin | Question: What about etale (and other) analogues of this? | |
| Oct 16, 2012 at 16:44 | answer | added | Ronnie Brown | timeline score: 13 | |
| Oct 16, 2012 at 11:45 | comment | added | Jeremy Brazas | Related: mathoverflow.net/questions/45351/does-pi-1-have-a-right-adjoint/… | |
| Oct 16, 2012 at 4:42 | answer | added | Marc Hoyois | timeline score: 40 | |
| Oct 16, 2012 at 4:25 | comment | added | Andrej Bauer | I always thought that the conditions of van Kampen theorem were about making sure that the diagram we start with is actually a pushout in the category of pointed spaces. But maybe I am wrong. | |
| Oct 16, 2012 at 4:02 | history | asked | ziggurism | CC BY-SA 3.0 |