The stable hull of an exact $\infty$-category

Abstract

We construct a left adjoint ${\mathcal{H}}^{\text{st}}:{\mathbf{Ex}}_{\infty }\to {\mathbf{St}}_{\infty }$ to the inclusion ${\mathbf{St}}_{\infty }\hookrightarrow {\mathbf{Ex}}_{\infty }$ of the $\infty$-category of stable $\infty$-categories into the $\infty$-category of exact $\infty$-categories, which we call the stable hull. For every exact $\infty$-category $\mathcal{E}$, the unit functor $\mathcal{E}\to {\mathcal{H}}^{\text{st}}(\mathcal{E})$ is fully faithful and preserves and reflects exact sequences. This provides an $\infty$-categorical variant of the Gabriel-Quillen embedding for ordinary exact categories. If $\mathcal{E}$ is an ordinary exact category, the stable hull ${\mathcal{H}}^{\text{st}}(\mathcal{E})$ is equivalent to the bounded derived $\infty$-category of $\mathcal{E}$.

22 pages, accepted version

References in corpus (4)

• Higher Topos Theory
• Stable Infinity Categories
• Controlled objects in left-exact $\infty$-categories and the Novikov conjecture
• Universal property of triangulated derivators via Keller's towers

Cited by in corpus (2)

• Positive and negative extensions in extriangulated categories
• On exact categories and their stable envelopes