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I have worked on the entry synthetic differential infinity-groupoid;
added a brief remark in the Idea section;
spelled out statement and proof that $SynthDiff \infty Grpd$ is totally $\infty$-connected over $Smooth \infty Grpd$;
began some discussion on how the induced relative fundamental $\infty$-groupoid functor is $\mathbf{\Pi}_{inf}$: the infinitesimal path $\infty$-groupoid functor, such that $\mathbf{\Pi}_{inf}(X)$ is the de Rham space of $X$ and a morphism $\mathbf{\Pi}_{inf}(X) \to \infty Mod$ an $\infty$-stack of D-modules on $X$. But this deserves more discussion.
Concerning the writeup of the second point I had myself confused about the direction of one of the arrows for a while. Hope I got it right now.
I have further refined the discussion. The point is that $Smooth\infty Grpd$ sits by what looks like a relative cohesive $\infty$-geometric morphism over $SynthDiff \infty Grpd$
$(i_! \dashv i^* \dashv i_+ \dashv i^!) : Smooth \infty Gprd \hookrightarrow SynthDiff \infty Grpd \,,$where however $i_!$ is full and faithful. This implies that with taking the lower three morphisms to regard conversely $SynDiff\infty Grpd$ as being strongly $\infty$-connected over $Smooth \infty Grpd$, the corresponding intrinsic relative fundamental $\infty$-groupoid functor
$\mathbf{\Pi}_{inf} := i_* i^*$has noth just the usual right adjoint $\mathbf{\flat}_{inf}$ for flat coefficients, but also a left adjoint for a total of
$(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf}) \,.$Looking at this in components one sees that this is precisely the setup for derived de Rham spaces as in Simpson-Teleman, only implemented here in geometry modeled on $C^\infty$-rings as opposed to other algebras.
In fact, I think this can be fully abstracted. By going through the details one can see that the key fact that encodes that these relative $\infty$-toposes exhibit an infinitesimal thickening is the coreflective embedding
$(i_! \dashv i^*) : Smooth \infty Grpd \hookrightarrow SynthDiff \infty Grpd \,.$Because here the right adjoint $i^*$ implements the “reduction” functor that quotients out nilpotent ideals and hence contracts away infinitesimal thickening, and the adjunction encodes that what is contracted away is indeed “infinitesimal” in that by
$Hom(i_! X, Y) \simeq Hom(X, i^* Y)$there is only a unique morphism from any non-thickened space $X$ into an infinitesimal point $i^* Y \simeq *$, saying that infinitesimally thickened point have only a unique global point, which is kind of the characterization of “infinitesimal”.
So I am now inclined to do the following, unless somebody tries to stop me: I’ll create an entry infinitesimally thickened cohesive (infinity,1)-topos (or maybe better formal cohesive (infinity,1)-topos?? or maybe infinitesimal cohesion??) where I make an quadruple of adjoijnts as above the general abstract definition of what it means to add infinitesimal cohesion to a cohesive topos, together with some discussion of how to interpret this and then with a pointer to synthetic differential infinity-groupoids as an example.
I have decided to put the general abstract discussion of infinitesimal objects into the list of structures at cohesive (infinity,1)-topos: in the new subsection Ininitesimal cohesion, Lie theory and deformation theory.
i think I have simplified a little – or at least streamlined – the proof at synthetic differential infinity-groupoid that $SynDiff\infty Grpd$ is an infinitesimal thickening of $Smooth \infty Grpd$
a little ? …oh I see it is about the early part of the sentence. Sorry.
Yes, sorry. :-) I have reordered the sentence now.
in the section “Cohomology” at synthetic differential infinity-groupoid I have written a discussion of its cohomology localization at the canonical line object. Needs polishing, but I have to interrupt now.
have started at synthetic differential infinity-groupoid a section Cohomology of oo-Lie algebroids
I have added statement and proof that a smooth function between smooth manifolds is a formally etale morphism in the general abstract sense with respect to the notion of cohesion of synthetic differential infinity-groupoids precisely if it is a local diffeomorphism in the traditional sense:
in this section
Great, this would be my first guess when restricted to smooth manifolds. Now what about the nonrepresentable morphisms of smooth spaces ?
Now what about the nonrepresentable morphisms of smooth spaces ?
Right, I should think about that. Another obvious next question is: is
formally smooth = submersion
formally unramifier = immersion
in this context? Locally certainly, but maybe globally not? I need to think more about this. It is all emenentary, but I need a more quiet moment to think it through carefully.
In case anyone else wants to join in, it’s just an analysis of the following simple type of pullback:
for $X$ any smooth space, $U$ a Cartesian space and $D$ an infinitesimally thickened point, so that $Hom(D,X)$ is the tangent space of $X$ for tangent vectors of shape $D$ (the ordinary tangent space when $D$ is the first order interval), we have the canonical morphism
$Hom(U \times D , X ) \to Hom(U , X)$that sends a smooth image of $U$ in $X$ equipped with a tangent field along $U$ to the underlying smooth map, forgetting the tangents.
Then for $f : X \to Y$ a morphism of smooth spaces, we say it is
formally étale,
formally unramified,
formally smooth,
respectively, if for all $U$, $D$ the square
$\array{ Hom(U \times D, X) &\stackrel{f_*}{\to}& Hom(U \times D, Y) \\ \downarrow && \downarrow \\ Hom(U , X) &\stackrel{f_*}{\to}& Hom(U , Y) }$is
a pullback
induces a monomorphism into the pullback $Hom(U,X) \times_{Hom(U,Y)} Hom(U \times D, Y)$
induces an epimorphism.
So it’s an elementary analysis. One just needs to concrentrate a bit.
Well, you talk immersions in the sense of differential geometry. In algebraic geometry, there is a notion of closed immersion and a different notion of an open immersion of schemes. The comorphism of open immersion of schemes is an isomorphism on stalks of the structure sheaf (basically by the definition). At the level of presheaves of sets, one has that open immersions are smooth (not only formally smooth) monomorphisms. The notion of immersion in differential geometry is somewhat different.
Yes, I mean differential geometry for the moment. I want to work out what the general abstract theory says in this case.
Somebody should also work out that the general abstract theory reproduces the setup in algebraic geometry. rosenberg-Kontsevich sort of do, but I feel that more details ought to be spelled out. But I don’t feel I can invest much time into that myself at the moment.
Somebody kindly reminds me by email that Anders Kock’s first book on SDG had a definition of formally étale maps in an SDG topos.
It is immediate to see that this coincides with the notion of formally étale maps in SynthDiff∞Grod as obtained from the general notion of diffential cohesion, when restricted to 0-truncated objects.
I made a quick note on this. Have to run now, will expand later.
I have added the argument and the pointer to Kock’s book to 4.5.6, p. 413 of dcct.
I have renamed “synthetic differential $\infty$-groupoid” – which was bad as a choice of terminology for a specific model – to formal smooth ∞-groupoid. Which is hopefully better.
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