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 is 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.



Based loop groups as stacks?
$\Omega G$ won't be a differentiable stack unless you are willing to go to infinite dimensions. Provided you are considering $S^1$ as a smooth manifold, Nerses answer above is correct- it gives you a nice sheaf on the site of manifolds (it's even concrete- a so-called diffeological space). It's also representable by an infinite dimensional manifold.


The other thing you could mean, is you might mean $S^1$ (or $B \mathbb{Z}$) as a homotopy type, or the "categorical circle." For this, you will have to go to derived manifolds (otherwise you will just recover $G$ back again), and then, $\Omega X = T[1] X$ is the shifted tangent bundle for any manifold $X,$ in particular, for $X=G$. $T[1] X$ is the graded manifold whose underlying space is $X$ and whose structure sheaf $\mathcal{O}$ is given by $\mathcal{O}^{-n}(U)=\Omega^{n}\_{dR}(U).$


*Edit*: Actually, in the above, I am computing $\mathcal{L} X$ the *free loops* on $X$:


For $x \in X,$ the based loops at $X$ are the homotopy fibered product $\* \times\_{X} T[1]X$, but since evaluation at the base point of $S^1$ is just the vector bundle map $T[1]X \to X,$ which is a submersion, the ordinary pullback is a homotopy pullback, so we get simply $\Omega\_x X=T\_x[1] X=\left(T\_x X\right)[1],$- the graded manifold associated to the graded vector space with $T\_x X$ sitting in degree $-1.$ It's underlying manifold is just a point, but it's structure sheaf is given by the exterior algebra of the dual this vector space (but placed in non-positive degrees).


