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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2014
    • (edited Jun 13th 2014)

    Gijs Heuts is visiting; today he gave an impressive talk about his ongoing work with Jacob Lurie on generalizing the story of rational homotopy theory to \infty-categories more general than Grpd\infty Grpd and to the case where the rational numbers are replaced by a Morava K-theory spectrum, hence the chromatic higher analogs of rational homotopy theory.

    This turns out to be encoded in Goodwillie-Taylor towers (higher Goodwillie jet theory). For one, the Goodwillie derivatives of the identity functor on Grpd *\infty Grpd_\ast together form the Lie \infty-operad in spectra (up to a degree shift).(!) After \mathbb{Q}-localization, this fact ends up implying the classical statement of rational homotopy theory, that connected rational spaces are equivalent to connective rational L L_\infty-algebras/dg-Lie algebras. Applying instead K(n)K(n)-localization it implies generalizations of rational homotopy theory to K(n)K(n)-local (unstable!) homotopy theory, making it similarly equivalent to L L_\infty-algebras in K(n)K(n)-local spectra.

    My question: From this perspective, what is fundamentally the reason that the Lie \infty-operad shows up here?

    Answer: First of all, actually more fundamental than the Lie operad is the co-commutative coalgebra \infty-co-operad, whose algebras come down to L L_\infty-algebras in nice enough situations. It’s really that co-operad which is fundamental to the story.

    My next question: Okay, so what is it that singles out the co-commutative coalgebra co-operad here?

    Answer: it’s the co-monad Σ Ω :SpectraSpectra\Sigma^\infty \Omega^\infty : Spectra \to Spectra

    (me thinking: hence the exponential modality of the linear homotopy-type theory of the given tangent \infty-category)

    … namely (roughly) after K(n)K(n)-localization the Σ Ω \Sigma^\infty \Omega^\infty co-monad becomes the co-monad associated to the co-operad of co-commutative co-algebras in K(n)K(n)-local spectra, and that is what ends up inducing the whole story.

    (Take that with a grain of salt as far as detailed statements are concerned. Maybe I’ll have a chance to recount more details later when I am awake, right now it’s late after dinner and beer…)

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 30th 2022

    If we ever dig into this again, Heuts explains what is mean by the co-commutative cooperad here, p. 12. It has the sphere spectrum as coefficients (p. 15).