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For any group $G$, the universal example for proper $G$-actions, $\underline{E}G$, is a proper $G$-space such that for any other proper $G$-space $X$, there exists a map (unique up to $G$-equivariant homotopy) $$X\to\underline{E}G.$$ The quotient $\underline{B}G$ may be called the classifying space for proper $G$-actions.

Question 1: Does a homomorphism of groups $f\colon G\to H$ induce a $G$-equivariant continuous map $f_*\colon\underline{E}G\to\underline{E}H$ and hence also a continuous map $\underline{B}G\to\underline{B}H$?

Question 2: Under what conditions do we have that $\underline{B}(G_1\times G_2)$ is homotopy equivalent/homeomorphic to $\underline{B}G_1\times\underline{B}G_2$?

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    $\begingroup$ 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. $\endgroup$
    – Kapil
    Commented Aug 13, 2021 at 2:00
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    $\begingroup$ 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". $\endgroup$
    – Konrad Waldorf
    Commented Aug 13, 2021 at 8:45
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    $\begingroup$ $\underline{B}G\not\simeq BG$ $\endgroup$
    – archipelago
    Commented Aug 13, 2021 at 15:42
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    $\begingroup$ 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. $\endgroup$
    – Fernando Muro
    Commented Aug 13, 2021 at 17:10
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    $\begingroup$ 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. $\endgroup$
    – Z. M
    Commented Aug 20, 2021 at 15:58

1 Answer 1

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Here is model that is obviously functorial: take for $\underline{E}G$ the simplicial complex with vertex set the finite subsets of $G$ and simplices the finite chains of sets ordered by inclusion. A homomorphism $f:G\rightarrow H$ induces a function from the finite subsets of $G$ to the finite subsets of $H$. You can also think of this model as the barycentric subdivision of the `simplex' with vertex set $G$.

I can't think of anything to add to Fernando Muro's answer to question 2 in the comments.

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