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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 nn-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

    Added the reference FelixHalperin

    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)

    diff, v81, current

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

    added full publication data for

    diff, v87, current

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