nLab orbit groupoid

If a group $G$ acts on a groupoid $\Gamma$, then there is an orbit groupoid $\Gamma /\! / G$ which is a groupoid with trivial $G$-action and with a $G$-morphism $\Gamma \to \Gamma /\! / G$ universal for $G$-morphisms to groupoids with trivial $G$-action. So in principle $\Gamma /\! / G$ is obtained from $\Gamma$ by identifying $g.\gamma$ and $\gamma$ for all $g \in G$, $\gamma \in \Gamma$.

Note that if a group $G$ acts on a space $X$ then it has an induced action on the fundamental groupoid $\Pi_1 X$, and so there is an induced morphism

$$\alpha: \Pi_1 X /\! / G \to \Pi_1 (X/G).$$

So there is interest in when this morphism is an isomorphism.

$\alpha$ is an isomorphism if $X$ is Hausdorff, has a universal cover, and the action of $G$ on $X$ is discontinuous.

This allows some calculation of the fundamental groups of orbit spaces.

References

• R. Brown, Topology and groupoids, Booksurge, 2006, Chapter 11.

• J. Taylor, “Quotients of groupoids by the action of a group, Math. Proc. Camb. Phil. Soc., 103 (1988) 239–249.