I'm 100% sure I read somewhere about a generic PBW theorem for the Lie operad, but I cannot find the reference. Anyone know where this is written down?
Hey @DenisNardin and @RuneHaugseng is there a place in HA that says localization (e.g. at a prime) preserves E_n algebras? Or that a localization functor of the module category lifts to one on algebras?
Maybe this is really easy when localization is smashing.
So, I cannot find it, but I think you can deduce it from the fact that the adjunction gives you an adjunction in the (∞,2)-category of ∞-operads, that is to say you have unit and counit that are lax symmetric monoidal natural transformations and that satisfy the triangular identity. Then they induce functors and natural transformations on the ∞-category of algebras satisfying the triangular identities
Hopefully there's a simpler way, but if not it shouldn't be too painful to write down
(if it's in HA it's probably buried somewhere in the section about operadic colimits)
Hm, so it seems like it's related to Example 7.3.2.8, which says that the adjunction lifts to LMod. Maybe it even follows from this because we can put algebras inside of LMod?
Ah, but I guess that might only be monoidal, at least as I've written it.