Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

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

Pages: 195 – 220

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a9


Jona Klemenc (Innsbruck, Austria)


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 $\mathcal{E}$.


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

Copyright © 2022, Jona Klemenc. Permission to copy for private use granted.

Received 13 November 2020

Received revised 5 September 2021

Accepted 8 September 2021

Published 10 August 2022