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1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then $$ Map(y(N),X) $$ is again a differentiable stack ($y$ is the Yoneda embedding). Assume that $X$ is represented by the Lie groupoid $(M_{0}, M_{1}, s,t)$. How to find a "good" Lie groupoid that represents $Map(y(N),X)$ (i.e., something that depend by $N, M_{0}, M_{1}, s$ and $t$)?

2) Assume that $X$ is a sheaves of simplicial sets over $Man$. If $X$ is $n$-geometric, then so is $Map(y(N),X)$. How to find a $n$-Lie groupoid that represents $Map(y(N),X)$?

Edit: As Dimitri Pavlov pointed out, it is not clear what does it means representability here. I'm looking for a simplicial possibly $\infty$ dimensional manifold (or simplicial diffeological space) $Z_{\bullet}$ such that

a) $y(Z_{\bullet})$ is a simplicial presheaf

b) the stackification of $y(Z_{\bullet})$ is weak equivalent to $Map(y(N),X)$

(hence for the case 1) I looking for a 2-truncated simplicial possibly $\infty$ dimensional manifold (or simplicial diffeological space) $Z_{\bullet}$, that enjoys the above properties.)

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    $\begingroup$ These mapping stacks are infinite dimensional (e.g., if N=S^1, X=S^1, you get the smooth loop space of S^1), so what do you mean when you say “a Lie groupoid that presents”? Do you allow infinite-dimensional Lie groupoids? $\endgroup$ Mar 17, 2016 at 11:03
  • $\begingroup$ Yes. But in order to have "representability", should I add this objects to $MAN$? By representability I mean the following: it is an infinite dimensional Lie groupoid (hence it defines a presheaves on simplicial sets) where stackifickation is equal to $Man (y(N), X)$ $\endgroup$
    – Cepu
    Mar 17, 2016 at 11:06
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    $\begingroup$ If you literally want the mapping stack to be a quotient of objects contained in the image of the Yoneda embedding, then the site Man must be drastically expanded. But in most applications (I'm not sure what your case is though) this is not necessary. $\endgroup$ Mar 17, 2016 at 11:47
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    $\begingroup$ The above description assumes the target is fibrant. Lupercio and Uribe do not require the target to be fibrant. This means that to compute the derived internal hom from S^1 to the target, the target must be fibrantly replaced, which amounts to sheafification. By Verdier's hypercovering theorem, one can equivalently cover the (representable) source by some cover W, then take the ordinary internal hom from the Čech nerve of W, and finally take the homotopy colimit over all W. This is what happens in their paper, though without modern tools and language the exposition is rather awkward. $\endgroup$ Mar 17, 2016 at 15:21
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    $\begingroup$ The only source that I know of where the Verdier hypercovering theorem is stated explicitly in its full generality is arxiv.org/abs/1502.03925v3. See also mathoverflow.net/questions/165302/… $\endgroup$ Mar 17, 2016 at 21:28

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