Let $G$ be a group (regarded as a one-object groupoid) acting on a groupoid $X$ (i.e.: a functor $F: G \to \mathsf{Groupoids}$ sending the unique object of $G$ to $X$). Denote with $X//G$ the “weak quotient” of such an action, i.e. the Grothendieck construction of the functor $F$.
It seems to be well-known that the homotopy cardinality of the weak quotient $X//G$ is given by the formula
$$
|X//G| = \frac{|X|}{o(G)}\text{,}
$$
where $|X|$ is the homotopy cardinality of $X$ and $o(G)$ is the order of $G$ (of course this makes sense only when $|X|$ and $o(G)$ are finite). However, I wasn't able to find any recorded proof of this fact (maybe it's very simple and I can't see it). Can someone sketch a proof or give some reference where I can find one?
Best Answer
If $p : E \to X$ is an $n$-sheeted covering of groupoids, then $|E| = n\cdot |X|$; apply this result to the case of the fibration of elements $p : X/\!\!/G \to G$, just reminding that that this map isn't a covering map unless the action of $G$ is free.
However, you can replace $p$ by a covering map $\tilde p$ that is sort of a fibrant replacement for $p$, in the category ${\bf Gpd}/X$; now $|G|=1/o(G)$ and $\tilde p$ has $|X|$ sheets!