Timeline for Reference request: functoriality of $\underline{E}$ and $\underline{B}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2021 at 1:20 | vote | accept | geometricK | ||
| Aug 20, 2021 at 15:58 | comment | added | Z. M | It seems to me that you are interested in the case that $G$ is endowed with a topology. If $G$ is a condensed group (given by, say, a Lie group), then $BG=[*/G]$ is a condensed anima and the usual "classifying space" is referring to its homotopy type. See Peter Scholze's Lecture. This point of view is heavily adopted in Fargues-Scholze. | |
| Aug 20, 2021 at 14:36 | answer | added | IJL | timeline score: 2 | |
| Aug 13, 2021 at 17:10 | comment | added | Fernando Muro | 1. Yes, you have different functorial constructions for both spaces. Not the one indicated above, though. 2. Yes, the product total space clearly satisfies the properties characterizing the universal space for proper actions so you get a homotopy equivalence. I don't know if you get a homeomorphism for some functorial construction, though. | |
| Aug 13, 2021 at 15:42 | comment | added | archipelago | $\underline{B}G\not\simeq BG$ | |
| Aug 13, 2021 at 8:45 | comment | added | Konrad Waldorf | This is model-dependent. If you construct $BG$ and $EG$ via geometric realization of the groupoids $G \Rightarrow \ast$ and $G \times G \Rightarrow G$, respecitvely, then everything should be fine. See Segal's paper "Classifying spaces and spectral sequences". | |
| Aug 13, 2021 at 6:56 | review | Close votes | |||
| Aug 18, 2021 at 3:03 | |||||
| Aug 13, 2021 at 2:00 | comment | added | Kapil | The constructions are not functorial. However, given $f:G\to H$ we can always replace $EG$ with $EG\times EH$ with the diagonal action. Now, there will be a projection map etc. | |
| Aug 13, 2021 at 0:52 | history | edited | geometricK | CC BY-SA 4.0 |
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| Aug 13, 2021 at 0:34 | history | edited | geometricK | CC BY-SA 4.0 |
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| Aug 13, 2021 at 0:06 | history | asked | geometricK | CC BY-SA 4.0 |