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Let $\textbf{CartSp}$ be the category of spaces of the from $\mathbb{R}^n$ with smooth maps between them. This is a site with respect to (differentially) good open covers, so consider the Grothendieck topos of sheaves of sets on this site.

Now $\textbf{CartSp}$ is not just a site but also the syntactic category of the Lawvere theory of smooth algebras, namely those $\mathbb{R}$-algebras $A$ for which every smooth map $\mathbb{R}^n \to \mathbb{R}^m$ lifts to a map $A^n \to A^m$ in a compatible way.

That is to say, a smooth algebra is a product preserving copresheaf on $\textbf{CartSp}$ and $\textbf{Sh}(\textbf{CartSp})$ is the classifying topos of smooth algebras.

Now let $X$ be an object of $\textbf{Sh}(\textbf{CartSp})$. Then there is a naturally defined sheaf of smooth algebras$$\mathcal{O}_X: \textbf{CartSp} \to \textbf{Sh}(\textbf{CartSp}) \to \textbf{Sh}(\textbf{CartSp})/X$$which is supposed to be regarded as the structure sheaf.

So the outlook of this is that the subcategory of concrete and locally representable sheaves on $\textbf{CartSp}$ is equivalent to the category of smooth manifolds.

The trouble I am having is figuring out how to think of these as smooth manifolds. It would seem that such a sheaf $X$ really does not have the structure of a smooth manifold since it does not come with a structure sheaf.

Evidently I should instead be thinking of the slice topos $\textbf{Sh}(\textbf{CartSp})/X$ it induces, since this comes with a structure sheaf as I described earlier.

But I can not seem to figure out how to view $\textbf{Sh}(\textbf{CartSp})/X$ as a "space" in its own right. What are its open sets?

The motivation for this question is that I am trying to learn étale machinery from an abstract nonsense perspective but for smooth manifolds instead of schemes.

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    $\begingroup$ I think you might be trying too many things as the same time. Are you comfortable with the notion of (Grothendieck) topos as a space-like thing? The "open sets" of a topos are the subobjects of the terminal object. This makes a lot more sense if you think of the topos associated to a topological space. $\endgroup$ – Denis Nardin Feb 1 '17 at 18:05
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    $\begingroup$ @DenisNardin I am familiar with space-like Grothendieck toposes. I'm good with open sets being subobjects of the terminal object, but I would like this to be restated in terms of probes since that would (hopefully) be more spatially intuitive $\endgroup$ – KeD Feb 1 '17 at 20:49
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You want to think about about the objects $X$ as you do the Yoneda image $h_M = Hom(-,M)$ of a manifold $M$, i.e., as sheaves on the site $\bf{CartSp}$

${\bf Sh}({\bf CartSp})/X$ is the "big little topos." Look at what it is--it is an over category: it's objects are (all the) arrows $U \to X$ with $U$ ranging in the big topos. This is essentially all of the information of how its peers may map into $X$.

A standard way to present manifold is via atlases/charts. We see a cover $\sqcup h_{U_i} \to X$ (or just a chart $h_{U_i} \to X$) are important objects in ${\bf Sh}({\bf CartSp})/X$. (Here if one wants, can take $U_i = \mathbb{R}^n \in Ob({\bf CartSp})$ [equal!].)

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