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

Jona Klemenc

Homology, homotopy, and applications, Tome 24 (2022) no. 2, pp. 195-220.

Résumé

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 $\mathcal{E}$.

Détail

DOI : 10.4310/HHA.2022.v24.n2.a9

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