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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2014
    • (edited Jun 10th 2014)

    Larusson formulates the Oka principle homotopy-theoretically as: a complex manifold XX is Oka if for every Stein manifold Σ\Sigma the canonical map

    Maps hol(Σ,X)Maps top(Σ,X) Maps_{hol}(\Sigma, X) \to Maps_{top}(\Sigma,X)

    between the mapping spaces is a weak homotopy equivalence (see here).

    It is natural to wonder what this looks like in terms of the cohesion of the \infty-topos AnlyticGrpd\mathbb{C}Anlytic\infty Grpd over CplxMfdCplxMfd.

    If we write Π:AnlyticGrpdGrpd\Pi : \mathbb{C}Anlytic\infty Grpd \to \infty Grpd, then up to possible technicalities to be checked, it should simply mean

    Π[Σ,X][ΠΣ,ΠX] \Pi[\Sigma, X] \stackrel{\simeq}{\longrightarrow} [\Pi \Sigma,\; \Pi X]

    where [,][-,-] is the internal hom.

    (Something close to this (but not quite the same) is what Lawvere calls the “axiom of continuity” in a cohesive topos.)

    If instead we work internally and let Π:AnlyticGrpdAnlyticGrpd\Pi : \mathbb{C}Anlytic\infty Grpd \to\mathbb{C}Anlytic\infty Grpd be the shape modality, then the above is equivalently

    Π[Σ,X][ΠΣ,ΠX]. \Pi[\Sigma, X] \stackrel{\simeq}{\longrightarrow} \flat [\Pi \Sigma,\; \Pi X] \,.

    In either case, it is a very natural condition to ask for in general cohesive \infty-toposes. Maybe one should call it the Oka-Larusson property or something…

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