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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.
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.
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)].
Thanks Dmitry, that’s great.
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