Quotient by the action of a group commuting with sequential colimits

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.

Solution 1:

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.)