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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 infinitygroupoid – 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.
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.)


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