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

We construct a left adjoint $\mathcal{H}^\textrm{st} : \mathbf{Ex}_\infty \to \mathbf{St}_\infty$ 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 $\mathcal{E} \to \mathcal{H}^\textrm{st} (\mathcal{E})$ 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 $\mathcal{H}^\textrm{st} (\mathcal{E})$ is equivalent to the bounded derived $\infty$-category of $\mathcal{E}$.

Keywords

exact infinity-category, stable infinity-category, bounded derived infinity-category

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Received 13 November 2020

Received revised 5 September 2021

Accepted 8 September 2021

Published 10 August 2022