ScalingStacks

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.