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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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2022

    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.