--Hopefully this question does not dublicate another-- In [this][1] question Tom Goodwillie pointed out, that the 'atlas part' of the definition of a smooth manifold can be redefined in terms of sheaves. How is that done? I assume it is something along the line: "An atlas is a sheaf on the site of cartesian spaces (the site with objects $\mathbb{R}^n$), such that ..." Can someone clarify this point? [1]: http://mathoverflow.net/questions/88056/is-there-a-sheaf-theoretical-characterization-of-a-differentiable-manifold