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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2014
   • (edited Jun 13th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexGijs 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...)

   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…)

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 30th 2022
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIf we ever dig into this again, Heuts explains what is mean by the co-commutative cooperad [here, p. 12](https://people.math.rochester.edu/faculty/doug/otherpapers/heuts-survey.pdf). It has the sphere spectrum as coefficients (p. 15).

   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).