Just in case anyone happens upon this old discussion, to point out that
this issue is meanwhile being discussed in some detail as the *smooth Oka principle* here.

Thanks Dmitry, that’s great.

]]>Yes. According to Corollary 31.38 of the soon-to-be-on-arXiv
draft http://dmitripavlov.org/concordance.pdf
the functor Π (denoted there by the fraktur C)
can be computed using the formula in Proposition 25.5
as ΠF = hocolim_{n∈Δ^op} F(**Δ**^n), where **Δ** denotes extended simplices considered as smooth manifolds
and F denotes an arbitrary object of E.

For representable stacks M we immediately see that Π recovers the underlying homotopy type because ΠM is simply the smooth singular simplicial set of M.

Applying this to the map Π[M,N]→[ΠM,ΠN],
we can compute Π[M,N] as the simplicial set whose k-simplices are smooth maps **Δ**^k × M → N
and [ΠM,ΠN] as [Sing(M),Sing(N)].
Thus the problem is reduced to the well-known comparison result (probably from 1950s?)
for the space of smooth maps M→N and its homotopical cousin [Sing(M),Sing(N)].

I am not sure in generality. For what it’s worth, it looks to me like the analogous question in complex analytic cohesion is the “Oka principle”, in this incarnation.

]]>Here’s a question (possibly naive) I have about the smooth infinity topos, which I’ll call $E$. (I guess I could ask this on MathOverflow, but anyone who is likely to know is probably here.)

I’ll write $T$ for infinity-groupoids, and $\Pi\colon E\to T$ for the functor left adjoint to the one sending a space $X$ to the constant simplicial sheaf with value $X$. I’ll identify the category $Man$ of smooth manifolds with a full subcategory of $E$; thus, $\Pi$ sends a manifold to its homotopy type.

Being an infinity topos, $E$ has (derived) internal function objects, which I’ll write as $[X,Y]$. The question is: if $M$ and $N$ are manifolds, is $\Pi[M,N] \approx [\Pi M,\Pi N]$, the latter being the derived mapping space in $T$?

It’s certainly true if $M=\mathbb{R}^k$, since $[\mathbb{R}^k,N]$ is “$\mathbb{R}$-homotopy equivalent” to $N$, and $\Pi$ inverts $\mathbb{R}$-homotopy equivalence.

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