ScalingStacks

A.2.5 Localizations[00IF]

Given an \(\infty\)-category \(\mathcal C\) and a collection \(\mathbf W\) of morphisms in \(\mathcal C\) (often assumed to be those defining a subcategory of \(\mathcal C\)), the localization of \(\mathcal C\) at \(\mathbf W\) is the target of the initial functor \(\mathcal C\rightarrow\mathcal C[\mathbf W^{-1}]\) that carries all morphisms in \(\mathbf W\) to equivalences. As an extreme example, the \(\infty\)-groupoid completion of \(\mathcal C\) is its localization at all of its morphisms; this defines a left adjoint \(\mathrm{Cat}_\infty \xrightarrow{(-)^{\textup{gpd}}} \mathcal S\) to the inclusion. More generally, we can identify the localization at (the morphisms in) a subcategory \(\mathbf W\subseteq \mathcal C\) as the pushout

A special case of localization is given by a reflective localization adjunction, i.e. an adjunction in which the right adjoint is fully faithful. In this case, writing \(\mathbf W\subseteq \mathcal C\) for the subcategory of morphisms in \(\mathcal C\) that are carried to equivalences in \(\mathcal D\), the left adjoint witnesses \(\mathcal D\) as the localization \(\mathcal C[\mathbf W^{-1}]\). In this case, \(R\) can be characterized as the inclusion of the full subcategory of objects of \(\mathcal C\) that are local with respect to the morphisms in \(\mathbf W\), i.e. those \(c \in \mathcal C\) such that for every \(d \rightarrow e\) in \(\mathbf W\) the morphism \(\mathrm{Hom}_\mathcal C(d,c) \leftarrow \mathrm{Hom}_\mathcal C(e,c)\) is an equivalence [Lur09, Prop. 5.5.4.2].⁴⁶ Of course, dual remarks pertain to coreflective localization adjunctions, i.e. adjunctions in which the left adjoint is fully faithful.

Note that we might obtain the same localization even if we change the collection \(\mathbf W\). For instance, it is unnecessary to invert equivalences (since they are already invertible), and for any pair of composable morphisms \(f\) and \(g\) inverting any two of \(f\), \(g\), and \(gf\) automatically inverts the third (since equivalences have the two-out-of-three property). This observation plays a key role in the theory of accessible localizations of presentable \(\infty\)-categories (see Subsection A.7).