Is there any significance to Bousfield localization in the non-derived context?

Question

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:

1. There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically just means a reflective subcategory of $C$.

2. There is also a more restrictive usage (as when talking about spectra), which requires $C$ to be monoidal, and means a reflective subcategory where the class of maps being localized at is of the form $\{X \mid E \otimes X = 0\}$ for some fixed $E \in C$.

In this question, I'm interested in the more restrictive usage (2).

In this sense, Bousfield localization makes sense in either an ordinary monoidal category or in a monoidal $\infty$-category (for that matter, the more general usage makes sense in either an ordinary category or in an $\infty$-category). But it's typically only discussed in an $\infty$-categorical setting (e.g. in model categories or triangulated categories).

Question 0: Is there a good reason why Bousfield localization for ordinary categories (in sense (2)) is rarely discussed?

I think the answer may be "yes" because the behavior of Bousfield localization may be quite different in the two settings, and it seems somehow "better" in the $\infty$-categorical setting. But I'm not sure how to articulate this.

Here are two examples of what I mean:

1. $E = \mathbb Z/p$:

   • When $C = Ab$ is the (ordinary) category of abelian groups and $E = \mathbb Z / p$, the Bousfield localization consists of the abelian groups which have no nonzero infinitely $p$-divisible elements.

   • But when $C = D(Ab)$ is the $\infty$-category of chain complexes of abelian groups (localized at the quasi-isomorphisms) and $E = \mathbb Z/p$, the Bousfield localization consists of chain complexes whose homology is $p$-complete.

2. $E = \mathbb Z_{(p)}$:

   • When $C = Ab$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of abelian groups which are $\ell$-torsionfree for $\ell\neq p$.

   • When $C = D(Ab)$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of chain complexes whose homology is a $\mathbb Z_{(p)}$-module.

By "different behavior", I mean, to a first approximation, that even though $D(Ab)$ is "the natural $\infty$-categorical counterpart to $Ab$", in these cases it's not the case that the restriction of the $E$-Bousfield localization in $D(Ab)$ to $Ab$ coincides with the $E$-Bousfield localization in $Ab$ itself.

Part of the problem is that I'm not exactly sure what qualifies as "being in the $\infty$-categorical setting". After all, an ordinary category is in particular an $\infty$-category. But maybe for concreteness, I'll ask a slightly less vague version of the question:

Question 1: If $T$ is a tensor triangulated category with a $t$-structure, and $E \in T^{heart}$, is there any reason to think about the Bousfield localization of $T^{heart}$ at $E$ rather than the Bousfield localization of $T$ at $E$?

Answer 1

I'm right in the middle of a busy teaching week, so I'll give an off-the-cuff answer. One place where localizations of type (2) were discussed in ordinary category theory was Brian Day's Note on Monoidal Localisation, which characterizes such localizations in just the same way I characterized monoidal Bousfield localizations in my thesis Monoidal Bousfield Localizations and Algebras over Operads. I learned later that similar considerations appeared in Barwick's paper On left and right model categories and left and right Bousfield localizations, but I think it's an "if" rather than an "iff." Perhaps an answer to Question 0 is "because they have been characterized and are a special case of type (1) localizations for ordinary categories." I don't think the answer is "because they are differently behaved." My sense from reading Lurie is that localization in the $\infty$-categorical setting works essentially the same as in the model categorical setting, assuming presentability. So, you should have the same characterization from my thesis in the $\infty$-categorical setting.

It's worth noting (for future readers) that the reason we take localizations of type (2) in algebraic topology is that Bousfield's original papers were about inverting homology theories $E$, and the $E_*(-)$ equivalences can be obtained via a type (2) localization, viewing $E$ as an object in the category of spectra. Note that you are not guaranteed in any of the settings that localizations play nicely with the monoidal product. In spectra, the monoidal localizations are exactly the stable localizations (also studied by Barnes and Roitzheim). A non-monoidal localization is given by the Postnikov section, an example I learned from Carles Casacuberta, and recounted in Localization of algebras over coloured operads. The problem with this example is on the homotopy level, so it affects all three settings.

Monoidal Bousfield localizations of tensor triangulated categories have been considered by Balmer and Sanders, e.g. on page 15 of The Spectrum of the Equivariant stable homotopy category of a finite group, and among other places I am sure. I remember Roy Joshua has also thought about the interplay of Bousfield localization, t-structures, and hearts. I have not heard about localizations of $T^{Heart}$ rather than $T$. I'd love to see a characterization of when localization commutes with taking the heart.

Answer 2

I think that one of the reasons why these localizations are rarely discussed in ordinary categories is that their application in a derived context often relies on stability.

Generally Bousfield localization starts with a collection of maps $S$ that you want to make into equivalences, defines an object $Y$ to be local if $Hom(B,Y) \to Hom(A,Y)$ is an isomorphism for any $f:A \to B$ in $S$, and then asks for a reflective ocalization onto the class of local objects.

Given $E$, you can either

1. localize at the maps $A \to B$ such that $E \otimes A \to E \otimes B$ is an equivalence, or
2. localize at the maps $X \to 0$ such that $E \otimes X$ is trivial.

In a stable setting, these are equivalent because $A \to B$ is an $E$-equivalence if and only if $E \otimes (B/A)$ is trivial. However, in an unstable setting the localizations are quite different. My suspicion is that people are often implicitly interested in applications of type (1) and that's why there's less of an emphasis on the types of localizations that you're discussing.

As a remark, in the unstable (but still homotopical) setting we often do construct nullifications by Bousfield localizing with respect to maps $W \to *$. However, the class of $W$ we are nullifying often has no description using $E \otimes W$ for any $E$ because they are often described using certain generators or a homology theory.