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Let $(\mathcal{C},\mathcal{T})$ be a site and $\mathcal{X}$ a stack over it. Suppose $(\mathcal{C},\mathcal{T})$ subcanonical and hence for every object $U$ in $\mathcal{C}$ consider the associated stack $\overline{U}$ (here with stack I mean a category $\mathcal{X}$ fibred in groupoids over $\mathcal{C}$ with projection $p_{\mathcal{X}}:\mathcal{X}\to\mathcal{C}$ satisfying "sheaf-theoretic" gluing conditions). Objects in $\overline{U}$ are pairs $(X,\phi)$ where $X\in\mathcal{C}$ and $\phi:X\to U$. A morphism $(X_1,\phi_1)\to(X_2,\phi_2)$ is just a morphism $X_1\to X_2$ in $\mathcal{C}$ commuting with the $\phi_i$'s.

We say that a stack $\mathcal{X}$ is representable if it is equivalent to $\overline{U}$ for some $U\in\mathcal{C}$.

I was wondering if the following statement is right or not: given a stack $\mathcal{X}$ and objects $U,V\in\mathcal{C}$ with morphisms $F:\overline{U}\to\mathcal{X}$ and $G:\overline{V}\to\mathcal{X}$, then there exists an object $T\in\mathcal{C}$ such that the fibre product $\overline{U}\times_{F,\mathcal{X},G}\overline{V}$ is equivalent to $\overline{T}$.

In the realm of $C^{\infty}$-algebraic geometry (see http://arxiv.org/pdf/1001.0023v4.pdf ) I have met the following assertion: Let $\overline{\underline{*}}$ be the $C^{\infty}$-stack associated to the $C^{\infty}$-scheme $\rm{Spec}\,\mathbb{R}$ and $x:\overline{\underline{*}}\to\mathcal{X}$ a 1-morphism, then there exists a $C^{\infty}$-scheme $G$ such that $\overline{\underline{*}}\times_{x,\mathcal{X},x}\overline{\underline{*}}\simeq\overline{G}$ (Definition 8.20 in the link above). I have tried to see that this is true by a direct computation, taking as $G$ the $C^{\infty}$-scheme $p_{\mathcal{X}}(x(\underline{*},id_{\underline{*}})$, but I was not able to conclude. I am quite convinced that this is not the right choice. At the same time, my feeling is that this result is valid in general, but since I cannot solve even the particular case above, I am not sure.

Do you maybe can suggest a right choice of $G$ in the particular case or an alternative argument used to face this kind of problems? And also, is the first statement right in general?

Thank you very much!

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    $\begingroup$ The statement that you propose is a reformulation of the condition that the diagonal of $\mathcal{X}$ is representable (at least if $\mathcal{C}$ has fiber products?), and such representability generally does not hold. For instance, in a scheme theoretic context, a condition of this sort is part of what it means for a stack to be algebraic (i.e., an Artin stack); but even in the good case when the stack is algebraic its diagonal need not be representable by schemes (but is only representable by algebraic spaces), although your statement would require representability by schemes. $\endgroup$ Sep 13, 2015 at 19:24
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    $\begingroup$ For what concerns your actual question about the pdf you linked, it seems from Def. 8.3 that a $C^\infty$-stack by convention means a geometric stack, which in turn by Def. 7.18 implies that its diagonal is representable. In other words, if I haven't overlooked anything, in your context your proposed statement results from the imposed definitions and from a general exercise about fiber products that amounts to stacks.math.columbia.edu/tag/04Z1 $\endgroup$ Sep 13, 2015 at 19:33
  • $\begingroup$ Dear Kestutis, thank you very much for your comment. In particular thank you for suggesting the result on stack project (which is an exercise, but at least you need to know the statement). I haven't noticed the strict link between representability of the diagonal and my question, but now I have got it. Best, Claudio. $\endgroup$
    – User3773
    Sep 14, 2015 at 10:32

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