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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2012
    • (edited Feb 17th 2012)

    The last two days Stephan Spahn was visiting me, and we chatted a lot about étaleness in cohesive ∞-toposes.

    We found proofs that

    • for every notion of infinitesimal cohesive neighbourhood

      i:HH th i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}

      the total space projections of locally constant \infty-stacks are formally étale;

    • the formally étale morphisms with respect to any choice of infinitesimal cohesion satisfy all the axioms of axiomatic open maps (or rather their \infty-version, of course).

    (These are to be written up. Requires plenty of 3d iterated \infty-pullback diagrams which are hard to typeset).

    Recall – from synthetic differential infinity-groupoid – that for the infinitesimal cohesive neighbourhood

    i:i : Smooth∞Grpd \hookrightarrow SynthDiff∞Grpd

    the axiomatically formally étale morphisms between smooth manifolds are precisely the étale maps in the traditional sense.

    Motivated by all this, I finally see, I think, what the correct definition of cohesive étale ∞-groupoid is:

    simply: XHX \in \mathbf{H} is an étale cohesive \infty-groupoid if it admits an atlas X 0XX_0 \to X by a formally étale morphism in H\mathbf{H}.

    I have spelled out the proof now here that with this definition a Lie groupoid 𝒢\mathcal{G} is an étale groupoid in the traditional sense, precisely if it is cohesively étale when regarded as an object of the infinitesimal cohesive neighbourhood SmoothGrpdSynthDiffGrpdSmooth \infty Grpd \hookrightarrow SynthDiff \infty Grpd.

    I hope to further expand on all this with Stephan. But I may be absorbed with other things. Next week I am in Goettingen, busy with a seminar on L L_\infty-connections.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2012
    • (edited Feb 18th 2012)

    I have added here a technical lemma assumed in the discussion above: that

    j !:SmoothGrpdSynthGrpd j_! : Smooth \infty Grpd \to Synth\infty Grpd

    sends an object presented by a simplicial smooth manifold to the object presented by the same simplicial manifold (but now in the synthetic \infty-topos).

    (Well, actually currently my proof needs that the simplicial manifold admits a simplicially compatible system of degreewise good open covers. That should always exist under mild conditions, but I think there is a proof for this on the nnLab only for nerves of Lie groups.)

    • CommentRowNumber3.
    • CommentAuthorStephan A Spahn
    • CommentTimeFeb 24th 2012
    • (edited Feb 25th 2012)

    -

    • CommentRowNumber4.
    • CommentAuthorStephan A Spahn
    • CommentTimeFeb 24th 2012
    • (edited Feb 24th 2012)

    -

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