Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?



Homotopy type of the simplicial action groupoid
Take the nerve of $X//G \in Gpd(sSet)$ to get a bisimplicial set (<http://ncatlab.org/nlab/show/bisimplicial+set>), then take the diagonal, or the Artin-Mazur codiagonal (<http://ncatlab.org/nlab/show/codiagonal>). Either of the resulting simplicial sets represent the homotopy type. Pick the one which is better for what you need it for.


