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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 ∞-categories more general than ∞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* together form the Lie ∞-operad in spectra (up to a degree shift).(!) After ℚ-localization, this fact ends up implying the classical statement of rational homotopy theory, that connected rational spaces are equivalent to connective rational L∞-algebras/dg-Lie algebras. Applying instead K(n)-localization it implies generalizations of rational homotopy theory to K(n)-local (unstable!) homotopy theory, making it similarly equivalent to L∞-algebras in K(n)-local spectra.
My question: From this perspective, what is fundamentally the reason that the Lie ∞-operad shows up here?
Answer: First of all, actually more fundamental than the Lie operad is the co-commutative coalgebra ∞-co-operad, whose algebras come down to L∞-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∞:Spectra→Spectra…
(me thinking: hence the exponential modality of the linear homotopy-type theory of the given tangent ∞-category)
… namely (roughly) after K(n)-localization the Σ∞Ω∞ co-monad becomes the co-monad associated to the co-operad of co-commutative co-algebras in 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…)
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).
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