ScalingStacks

A.8.9 Localizations of \(\mathcal O\)-monoidal \(\infty\)-categories[00IZ]

Given an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)\) and a collection \(\mathbf W\) of morphisms in \(\mathcal C\), the \(\mathcal O\)-monoidal localization of \(\mathcal C\) at \(\mathbf W\) is (the target of) the initial object of \(\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)_{\mathcal C/}\) in which the morphisms in \(\mathbf W\) are sent to equivalences. Of course, this generalizes the notion of localization of \(\infty\)-categories discussed in Subsection A.2.5.

As an important special case, we say that a reflective localization ([00IG]) is compatible with a (symmetric) monoidal structure \(\otimes \coloneqq \otimes^\mathcal C\) on \(\mathcal C\) if for all objects \(c,c' \in \mathcal C\) the morphism \(L(c \otimes c') \xrightarrow{L(\eta_c \otimes \eta_{c'})} L(RL(c) \otimes RL(c'))\) in \(\mathcal D\) is an equivalence.⁵⁹ In this case, \(\mathcal D\) inherits a (resp. symmetric) monoidal structure \(\otimes^\mathcal D\), defined by the formula \(d \otimes^\mathcal Dd' \coloneqq L(R(d) \otimes^\mathcal CR(d'))\) for any \(d,d' \in \mathcal D\) and with unit object \(\mathbbm{1}_\mathcal D\coloneqq L(\mathbbm{1}_\mathcal C)\),⁶⁰ and the left adjoint \(L\) is canonically (resp. symmetric) monoidal (so that the right adjoint \(R\) is canonically laxly (resp. symmetric) monoidal). In this case, the left adjoint \(L\) witnesses \(\mathcal D\) as not just a localization but also a (resp. symmetric) monoidal localization of \(\mathcal C\).