Timeline for Compelling evidence that two basepoints are better than one
Current License: CC BY-SA 2.5
9 events
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| Oct 4, 2010 at 12:49 | comment | added | Daniel Moskovich | I've thought about this some more, and this well-written answer makes a lot of sense. But is this approach really all that much simpler and more natural than what we usually do, which is to work with covering spaces, and to mod out later by the indeterminancy induced by choosing a basepoint? | |
| Oct 3, 2010 at 21:36 | comment | added | Ryan Budney | What's the relationship between the classifying space of a groupoid and its subgroups? It seems like there should be a fibration linking the two. So the reduction to groups in general seems to involve some effort. | |
| Oct 3, 2010 at 21:27 | comment | added | Harry Gindi | At least a convincing argument I've heard is that by fixing a basepoint and throwing out the rest of the data, we're losing actual information about the space. It's only okay to pick a representative if you can show that any other representative will not only be isomorphic but uniquely so. To put it in topological terms (and in a way made precise by looking at simplicial localizations), we can only pick a basepoint without throwing away information if the groupoid is contractible. | |
| Oct 3, 2010 at 21:10 | history | edited | André Henriques | CC BY-SA 2.5 |
In my example, the isotropy group is $A_n$, not $S_n$.
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| Oct 3, 2010 at 21:06 | comment | added | André Henriques | The notion of groupoid is closed enough to the notion of group that I don't expect that there are any results that really require groupoids in their proofs. In other words, if you're willing to pick a base points in every connected component of every groupoid that you encounter, then you'll be able to translate any argument into the world of groups. | |
| Oct 3, 2010 at 20:47 | comment | added | Ryan Budney | In low-dimensional topology, rather than using the terminology orbifold people would just talk about marked surfaces or manifolds with trivalent graphs embedded, etc. Orbifolds (and your groupoid formalism) would have been talked about as a type of labelled stratified space. This is symbiotic with the graph-of-groups terminology, described below. So the natural question to ask is, what do you "gain" by using groupoid terminology instead of more old-fashioned terminology -- other than being fashionable? I think that's why Daniel is asking about new theorems, etc. | |
| Oct 3, 2010 at 20:11 | history | edited | André Henriques | CC BY-SA 2.5 |
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| Oct 3, 2010 at 20:06 | history | edited | André Henriques | CC BY-SA 2.5 |
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| Oct 3, 2010 at 19:32 | history | answered | André Henriques | CC BY-SA 2.5 |