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5:56 AM
@TomBachmann This looks like it implies the result I was looking for thank you!
 
6:43 AM
If anyone is in the netherlands next week, join us: meta.mathoverflow.net/questions/4270/…
for the mathoverflow celebration (or at any of the other locations!)
 
 
8 hours later…
2:48 PM
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?
 
2:58 PM
@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 :).
 
 
4 hours later…
6:57 PM
@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.
 

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