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

TL;DR

This paper constructs a functor called the stable hull that embeds exact $ abla$-categories into stable $ abla$-categories, generalizing the Gabriel-Quillen embedding to the $ abla$-categorical setting, and recovers derived categories for ordinary exact categories.

Contribution

It introduces the stable hull as a left adjoint to the inclusion of stable $ abla$-categories into exact $ abla$-categories, extending classical embeddings to the $ abla$-categorical context.

Findings

The stable hull functor is fully faithful and preserves exact sequences.

For ordinary exact categories, the stable hull recovers the bounded derived $ abla$-category.

Provides an $ abla$-categorical analogue of the Gabriel-Quillen embedding.

Abstract

We construct a left adjoint $\mathcal{H}^\text{st}\colon \mathbf{Ex}_{\infty} \rightarrow \mathbf{St}_{\infty}$ 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 $\mathcal{E} \rightarrow \mathcal{H}^\text{st}(\mathcal{E})$ 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 $\mathcal{E}$.