Localizations of model categories and $\infty$-categories

Question

I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories.

According to Hirschhorn's book we can form the left Bousfield localization of a left proper cellular model category along any set of maps. According to Lurie's book we can form the (left) localization of a presentable $\infty$-category along a small collection of maps. How are those two results related? Is the simplicial nerve of a left proper, cellular model category a presentable quasi-category?

Furthermore we also know that we can form the right Bousfield localization of a right proper cellular model category along any set of objects. Is there an analogous theory of right localizations on $\infty$-categories?

EDIT: It has been pointed out that combinatorial model categories correspond exactly to presentable $\infty$-categories and left Bousfield localizations exist for every left proper, simplicial, combinatorial model category and correspond exactly to the localizations of the corresponding presentable $\infty$-category.

This still leaves the question if there is a similar theory and an existence theorem for right localizations of $\infty$-categories corresponding to the theory of right Bousfield localizations of something like right proper, simplicial, cellular model categories.

Thanks already!

Answer 1

There is also an existence theorem for right Bousfield localizations of presentable $\infty$-categories. In fact, it follows from the existence theorem for left Bousfield localizations.

Let $K$ be a collection of objects in $C$ and let $D\subset C$ be the subcategory of $K$-colocal objects. Then $D$ is manifestly closed under colimits. By the adjoint functor theorem, the inclusion $D\subset C$ has a right adjoint provided that $D$ is accessible. I claim that $D$ is accessible if the colimit closure $\bar K$ of $K$ in $C$ is accessible, for instance if $K$ is small.

To prove this, let $W$ be the class of $K$-colocal equivalences, viewed as a full subcategory of $C^{\Delta^1}$. Note that $W$ is the class of $E$-local objects for

$$ E=\{(0\to x) \to (x\stackrel=\to x),\quad x\in \bar K_0 \}, $$

where $\bar K_0$ is small and generates $\bar K$ under colimits. By the existence theorem for left Bousfield localizations (HTT 5.5.4.15), $W$ is presentable. Consider the functor $F: C\to (\mathrm{Fun}^R(W,S)^{op})^{\Delta^1}\simeq W^{\Delta^1}$ which sends $c\in C$ and $f:a\to b$ in $W$ to $\mathrm{Map}(c,a)\to \mathrm{Map}(c,b)$. The functor $F$ is accessible since it preserves colimits. The inclusion $D\subset C$ is the homotopy pullback by $F$ of the diagonal $W\to W^{\Delta^1}$. By HTT 5.4.6.6, $D$ is accessible.

We can be a bit more precise. Unraveling the previous argument, we see that the right adjoint $R: C\to D$ is computed as follows. The counit $Rc\to c$ is $L(0\to c)$ where $L$ is the left adjoint of $W\subset C^{\Delta^1}$. The usual construction of $L$ by transfinite induction shows that in fact $Rc\in\bar K$, so that $D=\bar K$. So if $C$ is $\kappa$-accessible and every object of $K$ is $\kappa$-compact, then $D$ is $\kappa$-accessible.

Answer 2

Though I don't know of an example off-hand, I don't believe it's the case that every cellular model category presents a presentable $\infty$-category. In practice, we can usually find a combinatorial model for the homotopy theory we're interested in.

On the other hand, I think it is true that if you start with a left proper, simplicial, cellular model category and localize, then you will get a localization of the corresponding $\infty$-categories. This is because of Proposition 5.2.4.6 in Higher Topos Theory, which says that a nice adjunction between simplicial model categories gives rise to an adjunction between $\infty$-categories.

You can probably remove the requirement that the model categories are simplicial if you work a bit harder.