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--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?

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    $\begingroup$ I think what Tom Goodwillie is referring to is the locally ringed space formalism. He seems to imply it when he says "Daniel likes the option of replacing (1) by the logically equivalent". You can also represent manifolds as sheaves on the site of cartesian spaces, but that is very different. $\endgroup$
    – Zhen Lin
    Commented Aug 13, 2014 at 13:33
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    $\begingroup$ I think this is common knowledge, for example you can check Ravi Vakil's book on the chapter on sheaves. $\endgroup$
    – Bombyx mori
    Commented Feb 26, 2015 at 6:45

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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.

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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.

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