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motivated by the blog discussion I added to rational homotopy theory a section Differential forms on topological spaces
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...
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 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 expanded the section on Sullivan models.
Probably eventually this should be split off into a separate entry.
Ah, no!! I accidentally erased it all…
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.
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.)
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.
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.
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:
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.
Added the reference FelixHalperin
Yves Félix and Steve Halperin, Rational homotopy theory via Sullivan models: a survey, arXiv:1708.05245
added pointer to Buijs-Murillo 12 (dg-models for non-connected rational spaces)
added full publication data for
added publication data for
added pointer to:
added these pointers on generalizing RHT to arbitrary fundamental groups:
Antonio Gómez-Tato, Stephen Halperin, Daniel Tanré, Rational Homotopy Theory for Non-Simply Connected Spaces, Transactions of the American Mathematical Society, Transactions of the American Mathematical Society Vol. 352, No. 4 (Apr., 2000), pp. 1493-1525 (33 pages) (jstor:118074)
Syunji Moriya, Rational homotopy theory and differential graded category, Journal of Pure and Applied Algebra, Volume 214, Issue 4, April 2010, Pages 422-439 (doi:10.1016/j.jpaa.2009.06.015)
Urtzi Buijs, Yves Félix, Aniceto Murillo, Daniel Tanré, Homotopy theory of complete Lie algebras and Lie models of simplicial sets, Journal of Topology (2018) 799-825 (arXiv:1601.05331, doi:10.1112/topo.12073)
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