The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage, which $\infty$-categorically just means a reflective subcategory of $C$.
There is also a more restrictive usage, 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 (as discussed in nLab) or in a monoidal $\infty$-category. But it's typically only discussed in an $\infty$-categorical setting.
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 $E$?
