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    • CommentRowNumber1.
    • CommentAuthorCharles Rezk
    • CommentTimeNov 17th 2015

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

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

    Being an infinity topos, EE has (derived) internal function objects, which I’ll write as [X,Y][X,Y]. The question is: if MM and NN are manifolds, is Π[M,N][ΠM,ΠN]\Pi[M,N] \approx [\Pi M,\Pi N], the latter being the derived mapping space in TT?

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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2015
    • (edited Nov 17th 2015)

    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.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 17th 2015

    Yes. According to Corollary 31.38 of the soon-to-be-on-arXiv draft 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)].

    • CommentRowNumber4.
    • CommentAuthorCharles Rezk
    • CommentTimeNov 17th 2015

    Thanks Dmitry, that’s great.