Timeline for Space of algebraic closures of $\mathbb{Q}$

6 events

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Apr 12, 2019 at 15:50	review	Suggested edits
Apr 12, 2019 at 16:30
Apr 12, 2019 at 9:17	comment	added	Maxime Ramzi		To answer your last question, I don't know what "crucial" means here, but there is a groupoid version of the Seifert-Van Kampen theorem, which allows one to compute the fundamental group of $S^1$, while the group version doesn't allow that. Similarly, you can compute the fundamental group of a quotient through groupoid techniques, and you can't always with the group. See Brown's Topology and groupoids
Apr 12, 2019 at 7:19	comment	added	Denis Nardin		As a more advanced version of what Achim Krause said, you can also take the profinite space given by the étale homotopy type of $\mathrm{Spec}\,k$ (which is also a $K(G_\mathbb{Q},1)$, although in a slightly more refined way that remembers the topology of $G_\mathbb{Q}$)
Apr 12, 2019 at 6:20	comment	added	Achim Krause		The collection of all algebraic closures naturally forms a groupoid, to get a "space of algebraic closures" you could take the nerve of that. Since the groupoid is connected, that is going to give you a $K(G_\mathbb{Q}, 1)$.
Apr 12, 2019 at 5:50	review	First posts
Apr 12, 2019 at 5:52
Apr 12, 2019 at 5:48	history	asked	user138264	CC BY-SA 4.0