Localization of lax symmetric monoidal categories

Abstract

In this note, we explain in some detail how one can fiberwise localize a (co)lax symmetric monoidal infinity-category. This construction was tacitly used in Section 5 of our recent paper "On the equivalence of the Lurie's infinity-operads and dendroidal infinity-operads". Version 2: A stronger version of the result is proven. Given a locally cocartesian fibration $f:X\to B$ and a collection of marked arrows $X^{\circ }\subset f^{-1}(B^{eq})$ in $X$ closed under locally cocartesian liftings, we prove that the localization $\mathcal{L}(X,X^{\circ })\to B$ is also a locally cocartesian fibration whose fibers are localizations of the fibers of $f$. This result is applied to the description of localizations of lax symmetric monoidal categories.

11 pages