Verdier quotients of stable quasi-categories are localizations

TL;DR

This paper proves that Verdier quotients of stable quasi-categories are equivalent to localizations, extending classical results to the quasi-categorical setting and exploring their compatibility with monoidal structures.

Contribution

It establishes that Verdier quotients in stable quasi-categories are localizations, generalizing known triangulated category results to the quasi-categorical context.

Findings

Verdier quotients are localizations in stable quasi-categories

Compatibility of Verdier quotients with symmetric monoidal structures

Elementary results on quasi-categories related to dg categories and derived categories

Abstract

The Verdier quotient $\mathcal{T}/\mathcal{S}$ of a triangulated category $\mathcal{T}$ by a triangulated subcategory $\mathcal{S}$ is defined by a universal property with respect to triangulated functors out of $\mathcal{T}$. However, $\mathcal{T}/\mathcal{S}$ is in fact a localization of $\mathcal{T}$, i.e., it is obtained from $\mathcal{T}$ by formally inverting a class of morphisms. We establish the analogous result for small stable quasi-categories. As an application, we explore the compatibility of Verdier quotients with symmetric monoidal structures. In particular, we record a few useful elementary results on the quasi-categories associated with symmetric monoidal differential graded categories and derived categories of symmetric monoidal Abelian categories for which we were unable to locate proofs in the literature.