ScalingStacks

Given an $\infty$-category ${\mathcal{C}}$ and a suitable collection of morphisms $S$, one might hope to form the localization ${\mathcal{C}}[S^{-1}]$. This is by definition an $\infty$-category equipped with a functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}[S^{-1}]$ which satisfies the following universal property: for any other $\infty$-category ${\mathcal{D}}$, restriction along $f$ identities

as the full subcategory of $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})$ spanned by those functors which send the morphisms in $S$ to equivalences in ${\mathcal{D}}$. If $S$ is a proper set, then ${\mathcal{C}}[S^{-1}]$ exists in the same universe as ${\mathcal{C}}$; indeed, without loss of generality we may assume that $S$ contains all degenerate edges of the simplicial set ${\mathcal{C}}$, in which case ${\mathcal{C}}[S^{-1}]$ may be constructed as a fibrant replacement of $({\mathcal{C}},S)$ in the model category of marked simplicial sets.

Often in practice, however, $S$ is not small, and the existence of ${\mathcal{C}}[S^{-1}]$ (without passing to a higher universe) requires more delicate analysis. One standard method is to show that ${\mathcal{C}}$ is presentable and $S$ is (generated by) a small set of arrows in a certain sense: this is the theory of Bousfield localization, following Bousfield’s seminal work on the subject [18]. In this case we may identify the localization as the full subcategory of ${\mathcal{C}}$ spanned by the $S$-local objects.

In model categories, there is a well-developed theory of Bousfield localization (e.g., Hirschhorn’s comprehensive discussion in  [42], Goerss and Jardine’s treatment in the simplicial setting [37], or the exposition of Smith’s theory for combinatorial model categories in  [2]). Because localization is a central technical device in our work, in this section we provide a brief review of Lurie’s version of localization in the setting of presentable $\infty$-categories from [52, §5.2.7] and [52, §5.5.4].

Specifically, we say that a colimit preserving functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ of presentable $\infty$-categories ${\mathcal{C}}$ and ${\mathcal{D}}$ is a Bousfield localization if the right adjoint of $f$ (which exists by the adjoint functor theorem) is fully faithful [52, 5.2.7.2]. When the context is clear, we tend to abuse notation and simply refer to this as a localization. A useful observation is that this data induces an equivalence between ${\mathcal{D}}$ and a full subcategory of ${\mathcal{C}}$, called the subcategory of local objects. In fact, [52, 5.2.7.4] gives a useful criterion for an endofunctor $L\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ to be a localization. Specifically, the following are equivalent:

1. (i)

   There exists a functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ with a fully faithful right adjoint $g$ and an equivalence $g\circ f\simeq L$.

2. (ii)

   When regarded as a functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L{\mathcal{C}}$, $L$ is the left adjoint of the inclusion $L{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$.

3. (iii)

   There exists a natural transformation $\alpha\colon\id_{{\mathcal{C}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L$ such that for objects $X$ in ${\mathcal{C}}$, the morphisms $L(\alpha(X))$ and $\alpha(LX)$ are both equivalences.

Recall that a functor is accessible if it is $\kappa$-continuous (preserves $\kappa$-filtered colimits) for some sufficiently large regular cardinal $\kappa$ [52, 5.4.2.5]. A localization is accessible if $g$ or $L$ are accessible functors (equivalently, see [52, 5.5.1.2]) or the the essential image $L{\mathcal{C}}$ is an accessible subcategory.

Accessible localizations of presentable categories can be completely classified as follows. Recall from [52, 5.5.4] that associated to any set of arrows $S$ in a presentable $\infty$-category ${\mathcal{C}}$, the Bousfield localization $S^{-1}{\mathcal{C}}$ is equivalent to the ordinary localization ${\mathcal{C}}[T^{-1}]$ of ${\mathcal{C}}$ at the strongly saturated class $T$ generated by $S$ [52, 5.5.4.5]. In particular, many different sets $S$ can generate the same strongly saturated class $T$; they all define the same full subcategory $S^{-1}{\mathcal{C}}$ of ${\mathcal{C}}$ of $S$-local objects [52, 5.5.4.15], where $S$-local is defined in the standard fashion [52, 5.5.4.1]. An accessible localization of a presentable $\infty$-category is presentable. As the notation suggests, Bousfield localization is characterized by the following universal property [52, 5.5.4.20]: for any other presentable $\infty$-category ${\mathcal{D}}$, composition with $L$ induces a functor

which is fully faithful and whose essential image consists of those colimit-preserving functors which take elements of $S$ to equivalences.