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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 17th 2012
   • (edited Feb 17th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe 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 [[cohesive (infinity,1)-topos -- infinitesimal cohesion|infinitesimal cohesive neighbourhood]] $$ i : \mathbf{H} \hookrightarrow \mathbf{H}_{th} $$ the total space projections of locally constant $\infty$-stacks are [formally étale](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#FormallySmoothEtaleUnramified); * 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](http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos+--+infinitesimal+cohesion)_ 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](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#FormallySmoothEtaleUnramified) in $\mathbf{H}$. I have spelled out the proof now [here](http://ncatlab.org/nlab/show/synthetic+differential+infinity-groupoid#StructureSheaves) 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 18th 2012
   • (edited Feb 18th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added [here](http://ncatlab.org/nlab/show/synthetic+differential+infinity-groupoid#StructureSheaves) 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.)

   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.)

3. • CommentRowNumber3.
   • CommentAuthorStephan A Spahn
   • CommentTimeFeb 24th 2012
   • (edited Feb 25th 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex-

   -

4. • CommentRowNumber4.
   • CommentAuthorStephan A Spahn
   • CommentTimeFeb 24th 2012
   • (edited Feb 24th 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex-

   -