TITLE: Is there any significance to Bousfield localization in the non-derived context?
QUESTION [12 upvotes]: The term "Bousfield localization" of a category $C$ is used in roughly two different ways:

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

$E = \mathbb Z/p$:

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.

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

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 $E$?

REPLY [6 votes]: 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

localize at the maps $A \to B$ such that $E \otimes A \to E \otimes B$ is an equivalence, or 
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.