Timeline for Coverings of a space and coverings of a groupoid
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 21, 2018 at 11:42 | comment | added | Ronnie Brown | One should also mention the work of J. Brazas on "semicoverings" HHA 2012. | |
| Jan 20, 2018 at 11:02 | comment | added | Ronnie Brown | @QiaochuYuan My 1967 paper on van Kampen's theorem used a set of base points chosen according to the geometry of the situation, an idea supported by Grothendieck in his 1984 "Esquisse.." Section 2. Also the groupoid $\mathcal I$ with two objects $0,1$ and exactly one element $\iota:0 \to 1$ does well as a model of the unit interval, giving a homotopy theory for groupoids, and also is a generator for the category of groupoids, as is the integers for the category of groups For more on such uses, see for example arxiv: 1207.6404, published in Advances, 2014. | |
| Jan 17, 2018 at 1:12 | comment | added | Jeremy | @QiaochuYuan: as you mentioned, since there seems to be no canonical choice of basepoints, I was implicitly taking all the points. | |
| Jan 17, 2018 at 1:12 | comment | added | Jeremy | @RonnieBrown: thank you. That is the kind of answers I was ultimately expecting. Maybe a follow up would be: How far from a covering a $\pi_1$-covering could be? And to join Tim's answer: How far from a good category of coverings the category of $\pi_1$-coverings could be? | |
| Jan 12, 2018 at 20:24 | comment | added | Qiaochu Yuan | @Ronnie: generally I think of the fundamental groupoid as an object only well-defined up to equivalence, and I exploit the freedom to use any set of basepoints containing at least one basepoint in each path component to define it (e.g. for a path connected space I exploit my freedom to use exactly one basepoint to get the fundamental group back). But since the definition of covering map of groupoids is not invariant under equivalence I now have to make a particular choice of basepoints. I guess you want to pick the maximal choice given by every point. | |
| Jan 12, 2018 at 12:37 | comment | added | Ronnie Brown | In answer to a part of Jeremy's question we could define a map $p: Y \to X$ to be a $\pi_1$-fibration (covering) if $\pi_1(p)$ is a fibration (covering) of groupoids. In the fibration case this would give the exact sequences of the fibration in low dimensions, as given in T&G 7.2.9. | |
| Jan 12, 2018 at 12:28 | comment | added | Ronnie Brown | I do not understand the suggestion about base points in Qiaochu's answer as the fundamental groupoid does not have a base point chosen. In the book "Topology and Groupoids" Section 9.5, a local condition involving $\pi_1(p)$ is given on $X$ for $p:\widetilde{X} \to X$ to be a covering map. . | |
| Jan 10, 2018 at 20:49 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |