The defining problem of homotopy theory is that often when one localizes a nice category at a reasonable class of morphisms, the result is a very bad category. Does passing to the $\infty$ world fix this (so that homotopy theory becomes no different from ($\infty$)-category theory)?

A "nice" category is an accessible one, a reasonable class of morphisms is one which is accessible and accessibly embedded in the arrow category, and the localization is a category with the appropriate universal property. This all transports into the $\infty$ setting. To make this precise, one might consider the strict 2-category (actually, the link only talks about the strict 2-category of all quasicategories, we pass to a locally full sub-2-category) $\mathbf{Acc}$ of accessible $\infty$-categories and accessible $\infty$-functors (which I guess has PIE limits as in the ordinary case, including cotensors to define the arrow category), and ask whether this 2-category has co-inverters. This fails in the non-$\infty$-case -- weak equivalences of simplicial sets are accessible and accessibly embedded, but the co-inserter is the homotopy category, which is not accessible.

Or maybe there's a more $\infty$-way to put this, talking about codescent diagrams or something.

More generally, one could ask whether the 2-category of accessible $\infty$-categories has all co-PIE colimits. Of course, all of this may be asking too much -- after all, the 2-category of $\lambda$-accessible categories and $\lambda$-accessible functors should be a 2-category of *free* algebras for the free $\lambda$-filtered cocompletion 2-monad $\mathrm{Ind_\lambda}$ -- and one generally expects that free algebras won't be closed under colimit-like constructions like quotients (or localizations).

I suppose that if the $\infty$-category $\mathcal{C}$ being localized is locally presentable, then presenting $\mathcal{C}$ as a combinatorial model category, we should be able to use Jeff Smith's theorem to see that the localization is again presented by a combinatorial model structure. At least, this is so long as the localizing subcategory satisfies 2-out-of-3. But it would be nice to remove this technical hypothesis.