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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.

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  • $\begingroup$ Note to self: the conclusion seems to be that HTT 5.5.4.15 is the key to $\infty$-categorical localizations. It says that if you ask for the universal property of a localization in the category $Pres^L$ of presentable $\infty$-categories and left adjoints, then it's given by a reflection. This is obvious when you phrase it in terms of $Pres^R$ (where morphisms are right adjoints) instead. And this is also true in ordinary category theory. The added power comes because in $\infty$-categories, more of the interesting localizations respect colimits and so live in $Pres^L$. $\endgroup$ – Tim Campion Apr 2 '18 at 21:03
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Just a few points: first, if you localize at a subcategory $W\subset\mathcal{M}$ that doesn't satisfy 2-out-of-3, it doesn't really matter, in the sense that you'll really be localizing at the smallest supercategory $W\subset \overline{W}\subset\mathcal{M}$ which does satisfy 2-out-of-3 (called the saturation of $W$). After all, if we localize at $g$ and $gf$, we get $f^{-1}=(gf)^{-1}g$ (note that we really do need to localize at $g$ as well for this equation to hold).

Second, as long as you're talking about presentable $(\infty,1)$-categories and localizing at small classes of morphisms, the $\infty$-version of localization is really just reflecting into the full subcategory of $\overline{W}$-local objects -- i.e. it is a left adjoint, which means that it preserves colimits (so if they existed before, they still exist). This is Proposition 5.5.4.15 of Higher Topos Theory. Also, note that $\mathrm{Hom}(-,A\times B)\cong\mathrm{Hom}(-,A)\times\mathrm{Hom}(-,B)$, which implies that the product of local objects is local, so the localization functor preserves products (this is true of infinite products as well), and the equalizer of $A\rightrightarrows B$ is $\overline{W}$-local as long as $A$ is, so we have all small 1-limits. Limits of diagrams indexed by non-discrete $(\infty,1)$-categories probably work out as well, but I don't know a proof off the top of my head.

The cautionary point is that all of this discussion takes place in the context of $\mathbf{qCat_\infty}$, not $\mathbf{qCat_2}$. And also, localizations of $(\infty,1)$-categories that are not presentable are not necessarily so well-behaved.

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    $\begingroup$ Regarding your first paragraph: I'm worried that the saturation of an accessible, accessibly embedded subcategory of the arrow category might fail to be accessible / accessibly embedded, so I think it potentially does make a difference. $\endgroup$ – Tim Campion Jul 3 '16 at 15:30
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    $\begingroup$ Regarding your second paragraph: In ordinary category theory the word "localization" is often taken to mean taking a subcategory orthogonal to some objects, analogous to the sort of localization you describe. But I mean a localization in the broadest possible sense of universally inverting a class of morphisms, accessibly embedded in the arrow category. E.g. the construction of the homotopy category is a localization in the broad sense but not the narrow sense. Do you say that every localization in this broad sense is a reflective localization in the context of presentable $\infty$-categories? $\endgroup$ – Tim Campion Jul 3 '16 at 15:31
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    $\begingroup$ Yes, it is true that I have not fully answered your question because my response only applies in the context of presentable $(\infty,1)$-categories; however, what I have written holds for presentable $(\infty,1)$-categories: localization in the broad sense coincides with localization in the narrow sense in the presence of those hypotheses, and localization at $W$ really is the same as localization at its saturation, so questions of accessibility are unimportant in this context. I do not know the answer for non-presentable $(\infty,1)$-categories. $\endgroup$ – Kyle Ferendo Jul 3 '16 at 15:40
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    $\begingroup$ Yes, I agree! One feels that ``localization'' is much more naturally an $\infty$-categorical concept than a 1-categorical concept. Unfortunately, I don't know a reference for accessible rather than small classes of morphisms. Usually when I encounter a size issue, I try to see if I can move into a larger Grothendieck universe, so that whatever object I want to be small actually is small. But I don't think that solves all the problems that might arise here, since perhaps in that case the $(\infty,1)$-category in question would no longer be presentable? Size issues clearly aren't my specialty. $\endgroup$ – Kyle Ferendo Jul 3 '16 at 15:48
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    $\begingroup$ @TimCampion "accessible classes" as you call them, are always generated by a set. See Lemma 5.5.4.14 in HTT. $\endgroup$ – Dylan Wilson Jul 3 '16 at 16:03

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