@TomBachmann This looks like it implies the result I was looking for thank you!

for the mathoverflow celebration (or at any of the other locations!)

Given an accessible localization of a presheaf $\infty$-category $C \subset Psh(I)$ (where is small) then for the stabilization of $C$ we have $Stab(C) \subset Psh^{Sp}(I)$ and this inclusion must commute with the $\Omega^{\infty}$ functor for formal reasons (we say that $\Omega^{\infty}$ is levelwise). Under what extra conditions on the localization do we have that the corresponding beck chevalley map is an equivalence? equivalently when is $\Sigma^{\infty}$ levelwise?

@SaalHardali a sufficient condition is that local objects are closed under colimits. Not sure if this ever happens...

@TomBachmann Oh yeah, I was hoping for something less restrictive.

Hoping this is equivalent to some other categorical property

Maybe Goodwillie calculus is relevant here?

I don't know. I'll be happy to learn if someone chimes in :).

@asdq Formulas for mapping spaces in localisations with calculi of fractions can also be found in Cisinski's book "Higher Categories and Homotopical Algebra", Section 7.2.