Verdier quotients of stable quasi-categories are localizations

Brad Drew

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arXiv · 1511.08287

Verdier quotients of stable quasi-categories are localizations

Brad Drew

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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.

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Brad Drew. 2015-11-26. Verdier quotients of stable quasi-categories are localizations. https://arxiv.org/abs/1511.08287

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