I am reading the paper "[A universal characterization of higher algebraic K-theory](https://arxiv.org/abs/1001.2282)" by Blumberg, Gepner, and Tabuada, and I am stuck on Corollary 4.25:



> 
> …the fact that we have accessible localizations provides the following corollary about the
> structure of $\mathrm{Cat}\_\infty^\mathrm{ex}$ and $\mathrm{Cat}\_\infty^\mathrm{perf}$.
> 
> 
> 



> 
> Corollary 4.25. The ∞-categories $\mathrm{Cat}\_\infty^\mathrm{ex}$ and $\mathrm{Cat}\_\infty^\mathrm{perf}$ are compactly generated, complete, and cocomplete.
> 
> 
> 


Here, $\mathrm{Cat}\_\infty^\mathrm{ex}$ (resp. $\mathrm{Cat}\_\infty^\mathrm{perf}$) is the ∞-category of small stable ∞-categories (resp. of small idempotent-complete stable ∞-categories).


The "accessible localizations" they are referring to are Theorem 4.22 and Theorem 4.23, which state that $\mathrm{Cat}\_\infty^\mathrm{ex}$ and $\mathrm{Cat}\_\infty^\mathrm{perf}$ are accessible localizations of the ∞-category of small spectral categories.


My thought is as follows: Dugger's theorem shows that the ∞-category of small spectral categories can be modelled by a combinatorial simplicial model category, so it is presentable. Therefore $\mathrm{Cat}\_\infty^\mathrm{ex}$ and $\mathrm{Cat}\_\infty^\mathrm{perf}$ are presentable. However, I cannot see how to prove that they are $\omega$-accessible, i.e. compactly generated. Can anyone help me fill in this gap?



Compact generation of the infinity category of stable infinity categories
Proposition 4.20 in the same paper shows that the localization is in fact $\omega$-accessible, i.e. the fully faithful right adjoint in question preserves filtered colimits. This implies that the localized category is $\omega$-accessible, in particular compactly generated (because the category you're localizing from is compactly generated)


Alternatively, you can observe that the forgetful functor $Cat^{ex}\_\infty\to Cat\_\infty$ preserves filtered colimits and arbitrary limits, and that therefore its left adjoint preserves compacts. Because this forgetful functor is furthermore conservative, the claim also follows.


