7
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ See A.3.7.8 in Higher Topos Theory. $\endgroup$ – prefaisceau Dec 9 '14 at 17:56
  • 2
    $\begingroup$ @prefaisceau the OP is asking about the case where we start with a cellular model category, not a combinatorial one. In which case it's not at all clear that the underlying $\infty$-category is presentable. $\endgroup$ – Dylan Wilson Dec 9 '14 at 18:26
6
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you! The main part of my question is rather if there is a theory of right localizations of $\infty$-categories, though. As part of this I want to understand that if left proper, simplicial, cellular translates to presentable, what does right proper, simplicial, cellular translate to? $\endgroup$ – COhrt Dec 9 '14 at 18:48
  • 4
    $\begingroup$ I don't think that "presentable" is the translation of left proper, simplicial and cellular. Roughly speaking "presentable" is the translation of combinatorial. Left proper and simplicial are properties that in a certain sense you have for free for $\infty$-categories $\endgroup$ – Denis Nardin Dec 9 '14 at 19:13
  • $\begingroup$ I am not so sure that you can remove the requirement about being simplicial. This doesn't feel like the sort of situation where it's just a choice of framing. Rather, Lurie's nice adjunction you mention seems to need the simplicial hypothesis in a stronger way. I guess what I'm saying is: I'd love to see someone actually do that work to remove the simplicial hypothesis, because I think I'd learn something from it. $\endgroup$ – David White Jun 19 '15 at 22:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.