added full publication data for

- Kathryn Hess,
*Rational homotopy theory: a brief introduction*, contribution to*Summer School on Interactions between Homotopy Theory and Algebra*, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar LAvramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.),*Interactions between Homotopy Theory and Algebra*, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)

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

]]>Added the reference FelixHalperin

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

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*.

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:

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.

]]>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.

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.)

]]>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. ]]>

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.

]]>Ah, no!! I accidentally erased it all…

]]>I expanded the section on Sullivan models.

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

]]>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.

]]>I am wondering about the following:

there is a Quillen adjunction 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 of at all the cylinder projections is Quillen equivalent to .

Let be cofibrant replacement in this localized structure. Then we get the composite functor .

Looking at what this does on objects, it seems it should be related to the Sullivan-forms functor . Has anyone thought about this or seen other people think about it?

]]>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...

]]>motivated by the blog discussion I added to rational homotopy theory a section Differential forms on topological spaces

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