I have been stuck for some time, thinking about the following question.

Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which of dimension $-dim(G)$.

Can something similar be said for the based loop group $\Omega G$ of pointed maps from $S^1$ to $G$ ? 

I mean, is $\Omega G$ a differentiable stack and, if so, what is its dimension?

Best,

O.