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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 22nd 2009
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 22nd 2009
• (edited Dec 22nd 2009)

I see that there was an entry by Tim Porter, that I had forgotten about: differential forms on simplices. I put a link to that in the context at rational homotopy theory now.

I also edited that entry a bit: the first paragraph said that this is to be the first entry in a sequence of three, but as far as I can see Tim has since not followed up on this. So I removed his announcement (saved it at the bottom of the entry, actually). Also, I see that the entry doesn't actually say anything about polynomial forms so far...

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 22nd 2009
• (edited Dec 22nd 2009)

I am wondering about the following:

there is a Quillen adjunction $\Omega^\bullet : SPSh(Diff)^{loc} \to dgAlg^{op}$ that sends an oo-stack on the cat of manifolds to its deRham algebra.

Moreover, there is a theorem that says that the left Bousfield localization $SPSh(Diff)_{I}^{loc}$ of $SPSh(Diff)^{loc}$ at all the cylinder projections $X \times \mathbb{R} \to X$ is Quillen equivalent to $SSet$ .

Let $Q : SPSh(Diff)_I^{loc} \to SPSh(Diff)_I^{loc}$ be cofibrant replacement in this localized structure. Then we get the composite functor $SSet \stackrel{\simeq}{\to} SPSh(Diff)_I^{loc} \stackrel{Q}{\to} SPSh(Diff)_I^{loc} \stackrel{Id}{\to} SPSh(Diff)^{loc} \stackrel{\Omega^\bullet}{\to} dgAlg^{op}$ .

Looking at what this does on objects, it seems it should be related to the Sullivan-forms functor $SSet \to dgAlg^{op}$ . Has anyone thought about this or seen other people think about it?

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeDec 28th 2009
• (edited Dec 28th 2009)

I am a little confused about "diff forms on top spaces". The equivalence of cat of simplicial sets and of topological spaces is just as infty topoi, isn't it ? I mean the construction in Sullivan's word is eventually a construction in PL-world and for a topologists the world of say topological manifolds and the world of PL-manifolds is nontrivially different (and some people done a lot on documenting this difference) and this is not repaired by the infty machinery. What do you think ?

Another thing which could be of interest to discuss here is the business of D-modules. Namely the semialgebraic triangulations of semialgebraic sets play role in the theory of constructible sheaves and dualities in the theory of D-modules involving them. Regarding that this is also a non-smooth setup for generalizations of connections, there might be some common points in the theory. But here positive characteristics works also fine.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 21st 2010

I expanded the section on Sullivan models.

Probably eventually this should be split off into a separate entry.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 21st 2010

Ah, no!! I accidentally erased it all…

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 21st 2010

Phew. I got it back out of my browser’s cache. Luckily, that remembers all the content of the edit panes in the nLab edit pages.

Anyway, the new content that I put in is now at Sullivan model.

• CommentRowNumber8.
• CommentAuthorjim_stasheff
• CommentTimeJan 12th 2013
The paper on Deformations of Rational Homotopy Types
arXiv:1211.1647
that Mike and I posted has received only one substantial comment.
We'd like to submit it for pub but with any improvements suggested.
• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeJan 12th 2013
• (edited Jan 12th 2013)

That’s Mike Schlessinger. (By typing [Deformations of Rational Homotopy Types](http://arxiv.org/abs/1211.1647), you get a clickable link: Deformations of Rational Homotopy Types, provided that you choose a format which supports Markdown. I usually use Markdown+Itex.)

• CommentRowNumber10.
• CommentAuthorjim_stasheff
• CommentTimeJan 13th 2013
Thanks, Todd - I'm still somewhat illiterate, though not a luddite.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 14th 2016
• (edited Jan 14th 2016)

I have edited a little bit at rational homotopy theory (that whole entry needs a serious polishing and completion at some point):

• gave the rationalization adjunction in the Sullivan approach its own subsection, such as to make it easier to spot this key statement in the entry;

• expanded just a little there, but this deserves to be expanded further;

• started an Examples-section with the example of rational spheres.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJan 14th 2016
• (edited Jan 14th 2016)

And I have merged the section previously titled “Lie-theoretic models” into the Idea-section, for it just surveys the models that are then described in the following sections. Re-edited a little in the process.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJan 14th 2016

Ah, I see that the example of rational $n$-spheres was also requested at Sullivan model and at rational topological space. Therefore I now gave it its own dedicated entry and linked to from there:

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeFeb 15th 2017
• (edited Feb 15th 2017)

I would like to bring the entry rational homotopy theory into better shape. Today I have been expanding and streamlining the section on the Sullivan approach.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeOct 29th 2017

Yves Félix and Steve Halperin, Rational homotopy theory via Sullivan models: a survey, arXiv:1708.05245

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeApr 12th 2018

added pointer to Buijs-Murillo 12 (dg-models for non-connected rational spaces)

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJan 5th 2019
• (edited Jan 5th 2019)

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 19th 2020

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 23rd 2020

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeAug 23rd 2020