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Let $M$ and $N$ be smooth manifolds. The cartesian product $M\times N$ has a natural manifold structure. Moreover, $M$ and $N$ can be seen as locally ringed spaces over $\operatorname{Spec}\mathbb{R}$ and the fiber product (in this category) $M\times_{\operatorname{Spec}\mathbb{R}}N$ always exists.

I wonder if both products coincide or not. (However, I know that they don't if we take the fiber product on the category of ringed spaces. This is because, in this case, the structure sheaf of the product is the tensor product of the structure sheaves.)

What about more general limits? And other kinds of manifolds?

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    $\begingroup$ A construction of products of locally ringed spaces is described here by Hanno Becker. $\endgroup$
    – Tim Campion
    Commented Sep 3, 2021 at 14:38
  • $\begingroup$ Dear @TimCampion, I actually knew that construction and it inspired my question. But I don't see why it solves my doubt $\endgroup$
    – Gabriel
    Commented Sep 3, 2021 at 14:39
  • $\begingroup$ Oh for sure it's not an answer, that's why it was just a comment. Have you tried working out what Hanno's construction gives you in the case at hand? It probably simplifies to some extent at least. $\endgroup$
    – Tim Campion
    Commented Sep 3, 2021 at 14:41
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    $\begingroup$ I did, in fact. But I couldn't go very far, even in the underlying set. For example, the fiber product of two schemes over $\mathbb{R}$ does not have the cartesian product as underlying set. So we have to use somewhere that $M$ and $N$ are manifolds. I just don't know where. $\endgroup$
    – Gabriel
    Commented Sep 3, 2021 at 14:45
  • $\begingroup$ I don't believe that the germ $\mathcal O_{\mathbb R^2,0}$ could be obtained algebraically from the algebraic tensor product $\mathcal O_{\mathbb R,0}\otimes_{\mathbb R}\mathcal O_{\mathbb R,0}$. $\endgroup$
    – Z. M
    Commented Sep 3, 2021 at 16:07

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Yes if you think of manifolds as $C^\infty$-schemes.

(Joyces's book on the subject is probably the most accessible reference, but you could try the original paper by Dubuc.) You can see a hint of this fact in Moerdijk and Reyes' Models of infinitesimal analysis where they prove that the $C^\infty$-ring of smooth functions on the product $M\times N$ is the coproduct of the $C^\infty$-rings of functions on $M$ and on $N$. And Spec sends coproducts to products...

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