The stable hull of an exact \(\infty\)-category (Q2107542)

Every abelian category has a canonical structure of an ordinary exact category given by the class of all short exact sequences. Conversely, every ordinary small exact category admits an embedding into an abelian category with nice properties, which is what is called the Gabriel-Quillen embedding [\textit{R. W. Thomason} and \textit{T. Trobaugh}, Prog. Math. 88, 247--435 (1990; Zbl 0731.14001), Theorem A.7.1]. There is an alternative description of exact categories as extension-closed subcategories of abelian categories. From the standpoint of derived categories, one can naturally work with the structure of triangulated categories, which do not have good categorical properties. Lurie [\url{arXiv:math/0608228}] proposed an enhancement for triangulated categories, which is what are called \textit{stable \(\infty\)-categories}. Exact \(\infty\)-categories were introduced by \textit{C. Barwick} [Compos. Math. 151, No. 11, 2160--2186 (2015; Zbl 1333.19003)] as a generalization of ordinary exact categories in the sense of [\textit{D. Quillen}, Lect. Notes Math. 341, 85--147 (1973; Zbl 0292.18004)]. Small exact \(\infty\)-categories together with exact functors between them form an \(\infty\)-category \(\boldsymbol{EX}_{\infty}\) containing as a full subcategory both \begin{itemize} \item the category of ordinary small exact categories and exact functors between them, and \item the \(\infty\)-category \(\boldsymbol{St}_{\infty}\) of small stable \(\infty\)-categories\ and exact functors between them. \end{itemize} This paper constructs a functor \[ \mathcal{H}^{st}:\boldsymbol{EX}_{\infty}\rightarrow\boldsymbol{St}_{\infty} \] called the \textit{stable hull} functor, which is left adjoint to the inclusion \[ \boldsymbol{St}_{\infty}\hookrightarrow\boldsymbol{EX}_{\infty} \] For every exact \(\infty\)-category \(\mathcal{E}\), the unit functor \[ \mathcal{E}\rightarrow\mathcal{H}^{st}(\mathcal{E}) \] is fully faithful, preserving and reflecting exact sequences, which 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 \(\mathcal{H}^{st}(\mathcal{E})\)\ is equivalent to the bounded derived \(\infty\)-category of \(\mathcal{E}\).