Timeline for Terminology question on covering spaces
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2011 at 23:12 | answer | added | Mark Grant | timeline score: 3 | |
| Dec 16, 2010 at 1:04 | comment | added | David Roberts♦ | You could call the representation $\Pi_1(X) \to Set$ a 'transport functor', because it is analogous to parallel transport (where we have a unique flat connection - the structure group is, after all, a zero-dimensional Lie group). | |
| Dec 15, 2010 at 22:56 | vote | accept | Pavol S. | ||
| Dec 15, 2010 at 21:50 | answer | added | Sergey Melikhov | timeline score: 6 | |
| Dec 15, 2010 at 16:13 | comment | added | Pavol S. | @Ryan: here are 2 pedagogical reasons for giving "fool' covering spaces" independent life and name: 1. the fact that they are equivalent to ordinary covering spaces (if the base is nice) can be described by lifting the topology from $X$ to $Y$, which is an easy operation. 2. Using the classification of covering spaces (for nice base), van Kampen theorem (for nice spaces) is the trivial statement that if something is locally (in $X$) a covering space then it is a covering space. Without this classification and for arbitrary spaces it is the same locality statement for fool's covering spaces. | |
| Dec 15, 2010 at 14:47 | comment | added | Ryan Budney | I think it would be misleading to mistake the functor you're talking about with the associated covering space. Generally I'm not aware of a standard name for this process. I suppose I'd call it the monodromy classification of covering spaces or the action of $\pi_1$ on the fibre, or something like that -- I don't think this categorical perspective is much more than a "repackaging" of a classical theorem, so I just call these things by their classical names. | |
| Dec 15, 2010 at 14:26 | comment | added | Todd Trimble | @Tyler: yes, of course. I put it that way to elicit a response from Trial, who has answered my question satisfactorily. | |
| Dec 15, 2010 at 14:20 | comment | added | Tyler Lawson | @Todd: Some authors demand that the base and the covering space be connected as well. It depends on whether one wants to state the fundamental theorem in terms of subgroups of the fundamental group or in terms of a more functorial description as Trial indicates. | |
| Dec 15, 2010 at 14:20 | comment | added | Pavol S. | @Todd: "fool's" because the equivalence of "fool's" with ordinary is only true for nice spaces, and (given the level of the course) is considered as one of more difficult theorems. – Trial 5 | |
| Dec 15, 2010 at 14:15 | comment | added | Pavol S. | Sorry for being unclear. Any covering space is also a "fool's covering space". For a locally path-connected and semilocally 1-connected spaces the two notions are equivalent (this is considered a "difficult theorem" in the cours). In a fool's covering space the set $Y$ is just a set, with no topology (if it's uclear, take "functor $\Pi_1(X)\to Sets$ " as a definition of fool's covering space, and forget the other description). | |
| Dec 15, 2010 at 14:08 | comment | added | Todd Trimble | @Ryan: If $p: \hat{X} \to X$ is the universal covering space, then $\Pi_1(X)$ acts on $p$ in the category of etale spaces over $X$, and given a functor $F: \Pi_1(X) \to Set$, the corresponding covering space is obtained as the tensor product $p \otimes_{\Pi_1(X)} F$. @Trial: why "fool's"? If we relax the usual surjectivity condition of covering spaces and allow empty fibers, isn't the category of covering spaces over $X$ (for nice $X$) equivalent to the toppos of functors $\Pi_1(X) \to Set$? | |
| Dec 15, 2010 at 13:36 | comment | added | Ryan Budney | What exactly is the covering space among your "fool's covering spaces"? I'm confused as to why you're talking about sets rather than spaces and maps. | |
| Dec 15, 2010 at 13:31 | history | asked | Pavol S. | CC BY-SA 2.5 |