I don't believe they're equal but they're equivalent.

\o @JonathanBeardsley

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?

(this won't work arbitrarily, but should definitely work in char 0)

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.

@JonathanBeardsley Isn't enough that the localization functor is symmetric monoidal?

I mean, for every localization compatible with the symmetric monoidal structure (but all Bousfield localizations are)

@DenisNardin yes that would absolutely be enough.

and i guess this would follow for smashing localizations? or this is true of all Bousfield localizations?

For all localizations at a spectrum

Ah ok. Good to know, haha.

HA.2.2.1.9, you only need that X⊗- preserves L-equivalences for each X

Yeah, just found that as well. Awesome!

Thanks :)

So, just checking, we additionally should have that a symmetric monoidal left adjoints lifts to a left adjoint of algebras? Is that true?

I'm pretty sure it is true (because in this case both the left and right adjoints exist as map of operads) but let me hunt down a reference

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.

Actually, Remark 7.3.2.13

seems to be it

Ah, good catch

Haha, can I put "finding shit in Higher Algebra" on my CV?