Action Groupoid

Let $\alpha:G\to X\to X$ be a Group Action. The action groupoid $X\sslash_{\alpha}G$ is the following Groupoid:

• Objects are elements of $X$
• A morphism $x\to y$ is a pair of a group element $g$ and a proof that $\alpha(g,x)=y$.

In other words, $X\sslash_{\alpha}G$ is the Category of Elements of $\alpha$ when viewed as a Presheaf.

Note that there is a canonical functor $X\sslash_{\alpha}G\to\mathbf{B}G$ to the Delooping of $G$; this endows $X\sslash_{\alpha}G$ with the structure of a Discrete Cocartesian Fibration.

References

• https://ncatlab.org/nlab/show/action+groupoid