Timeline for Abstract definition of properly discontinuous action
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 11, 2014 at 21:23 | vote | accept | Earthliŋ | ||
| May 7, 2012 at 0:13 | answer | added | David Roberts♦ | timeline score: 3 | |
| May 6, 2012 at 8:45 | comment | added | Marc Palm | There hasn't been any improvement here. The question is now obvious. If you throw together en.wikipedia.org/wiki/Group_object and consider an action as the obvious thing, and require precisely what Fernando Muro writes above, you get a generalization. Let in the catagory of sets every objects carry the trivial toplogy, so compact set are finite sets there. Same for Lie groups, topological group, totally disconnected group... I am not sure about algebraic group schemes, since I do not know what proper morphisms are there, but they are probably chosen such that they do the same thing. | |
| May 6, 2012 at 7:26 | history | edited | Earthliŋ | CC BY-SA 3.0 |
extension of the reformulation
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| May 6, 2012 at 6:45 | history | edited | Earthliŋ | CC BY-SA 3.0 |
reformulation of the question
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| May 6, 2012 at 6:40 | comment | added | Earthliŋ | Thank you, Qiaochu Yuan, that is probably the best formulation of the answer I am looking for: How can the definition of a proper action be defined for group objects in an arbitrary category (such that it reduces to a proper action of a Lie group in the category of smooth manifolds and a properly discontinuous action of a group object in the category of sets)? | |
| May 4, 2012 at 17:47 | comment | added | Alain Valette | Indeed the question is not precise. | |
| May 4, 2012 at 17:44 | comment | added | Qiaochu Yuan | Presumably the motivation is to find a definition that generalizes to other categories (e.g. replacing groups with group objects). | |
| May 4, 2012 at 17:26 | history | edited | Misha | CC BY-SA 3.0 |
added compactness assumption for $K$
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| May 4, 2012 at 16:02 | comment | added | Martin Brandenburg | I agree that the question isn't precise at all. What is a "formal way of characterising a proper map" supposed to be, in contrast to one of the usual definitions? Besides, I've deleted some inappropriate tags. | |
| May 4, 2012 at 16:01 | history | edited | Martin Brandenburg |
edited tags
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| May 4, 2012 at 14:52 | comment | added | Marc Palm | The question is not really precise, since you do not say what particular property you like about properly discontinuous actions. Do you want to have a nice orbit/quotient space, then proper actions give you quotients with a Hausdorff topology. Moreover, you have a surjection $C_c(M) \twoheadrightarrow C_c(M)^G$, since the stabilizers are compact. | |
| May 4, 2012 at 12:53 | comment | added | Fernando Muro | A map is proper if the preimage of any compact subspace of the target is compact. Don't you think this is a formal enough characterization? You can't expect anything else from topology. Topology is defined in terms of families of subsets, you cannot avoid them. A group action of $G$ on $X$ is proper if the twisted diagonal $G\times X\longrightarrow X\times X\colon (g,x)\mapsto (gx,x)$ is proper. A discrete group action is properly discontinuous whenever it is proper in the previous sense. | |
| May 4, 2012 at 12:44 | history | asked | Earthliŋ | CC BY-SA 3.0 |