In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over $T\in GL^+(2,\mathbb{R})$ as the increasing maps $f:\mathbb{R}\to\mathbb{R}$ such that $f(r+1)=f(r)+1$ and $T$ and $f$ induce the same maps on $S^1$ when seen as a quotient of $\mathbb{R}$ or a subspace of $\mathbb{R}^2$ as appropriate.
Now, I can see that this does give a description of the appropriate covering space, since for every $T$ the fiber's really just $\mathbb{Z}$ ($f$ is determined by $T$ and by which element of $\mathbb{Z}+\theta$ it takes $0$ to, where $\theta$ is the argument of $T((1,0))$.) But I don't see how this is a particularly natural way to think about $G$. Is it an instance of a more general construction of covering groups, for instance? Or is Bridgeland just being very clever in writing $\mathbb{Z}$ so as to make his action easy to define?