Atlas of a manifold as a Sheaf --Hopefully this question does not dublicate another--
In this 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?
 A: 
An atlas is a sheaf on the site of cartesian spaces (the site with objects R   ), such that ...

One can certainly define smooth manifolds in such terms.
The cartesian site has finite-dimensional real vector spaces as objects, smooth maps between them as morphisms, and is equipped with the Grothendieck topology of jointly surjective families of submersions.
The category of sheaves of sets on this site will be henceforth referred to as the category of smooth sets.
A morphism of smooth sets is etale if its base change
along any morphism from a representable sheaf
is an etale morphism.
An atlas of a smooth set X is an etale epimorphism U→X
whose source is a coproduct of representables.
The category of smooth manifolds admits an obvious
fully faithful embedding into the category of smooth sets.
Its essential image coincides with smooth sets
that admit an atlas.
A: Section 4.1 of this paper by Dusko Pavlovic and Bertfried Fauser has a brief description of what I believe you're looking for, in the general situation of manifolds modelled on normed vector spaces. Section II.3 of Sheaves in Geometry and Logic by Moerdijk and MacLane discusses it in more detail for the usual case of finite-dimensional manifolds.
