Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 2 of 2
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…)
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).
1 to 2 of 2