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Let $G$ be a group and suppose $X_1 \to X_2 \to X_3 \to \cdots$ is a sequence of topological $G$-spaces and continuous $G$-maps. The colimit of this sequence inherits then a $G$-action.

Under what conditions does the colimit of the previous sequence commute with taking the quotient by the group action? That is, what conditions guarantee that $$ (\mathrm{colim} \ X_n)/G \cong (\mathrm{colim} \ X_n/G) ? $$

A naive example where this is true is $\mathbb{RP}^\infty = \mathrm{colim} \ \mathbb{RP}^n = S^\infty / (\mathbb{Z}/2)$. A more elaborate example (ultimately I am interested in this) is the Milnor construction of the classifying space for principal $G$-bundles, where (if I am not mistaken) $BG = EG/G$ but also $BG \cong \mathrm{colim} \ (E_n G/G)$ where $E_n G= G * \cdots *G$ is the $n$-fold join.

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This is always true; you can easily verify that the universal properties of the two spaces are equivalent. For instance, let $X=\operatorname{colim}X_n$. Then a map out of $X$ is the same as a compatible family of maps out of each $X_n$. To say that such a map is constant on $G$-orbits is equivalent to saying that the maps out of each $X_n$ are constant on $G$-orbits, and thus is equivalent to a compatible family of maps out of the spaces $X_n/G$. Thus $X/G$ has the universal property of $\operatorname{colim}X_n/G$.

(This is an example of the more general phenomenon that colimits commute with colimits, since the quotient by the group action is a type of colimit. It is only when you mix limits with colimits that you start needing extra conditions.)

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  • $\begingroup$ Very illustrative answer Eric! For the record: could you very briefly state what conditions you need to impose if you want such a sequencial colimit to commute with taking fixed points (which is a limit)? $\endgroup$
    – Minkowski
    Commented May 21, 2021 at 20:46
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    $\begingroup$ I don't know off the top of my head. There are a lot of weird things that could happen--for instance, you could have orbits that get smaller and smaller in each $X_n$ so that in the colimit they become fixed points that do not come from any fixed points in $X_n$. Or, even if that doesn't happen (so the colimit commutes with fixed points on the underlying sets), the topology on the colimit of the fixed points could be different from the topology on the fixed points of the colimit. $\endgroup$ Commented May 21, 2021 at 22:35
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    $\begingroup$ I guess it would suffice for the maps $X_n\to X_{n+1}$ to all be injective and for $X_n^G$ to be closed in $X_n$ for each $n$ (which is automatic if $X_n$ is Hausdorff). $\endgroup$ Commented May 21, 2021 at 22:39

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