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).

]]>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 $\infty Grpd$ 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 $\infty Grpd_\ast$ 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 $L_\infty$-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 $L_\infty$-algebras in $K(n)$-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 $L_\infty$-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 $\Sigma^\infty \Omega^\infty : Spectra \to Spectra$…

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

… namely (roughly) after $K(n)$-localization the $\Sigma^\infty \Omega^\infty$ 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…)

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