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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

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

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

$i :$ 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: $X \in \mathbf{H}$ is an étale cohesive $\infty$-groupoid if it admits an atlas $X_0 \to X$ by a formally étale morphism in $\mathbf{H}$.

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

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 $L_\infty$-connections.

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

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

$j_! : Smooth \infty Grpd \to Synth\infty Grpd$

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 $n$Lab only for nerves of Lie groups.)

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

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

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