The presentable stable envelope of an exact category

TL;DR

This paper extends classical embedding theorems to exact $$-categories, constructs a symmetric monoidal structure on their $$-categories, and explores implications for algebraic K-theory and synthetic spectra.

Contribution

It introduces a presentable stable envelope for exact $$-categories and develops a symmetric monoidal structure, linking to synthetic spectra and algebraic K-theory.

Findings

Established a Gabriel--Quillen type embedding for exact $$-categories.

Constructed a symmetric monoidal structure on the $$-category of small exact $$-categories.

Demonstrated the unique delooping of algebraic K-theory as a localising invariant.

Abstract

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal structure on the $\infty$-category of small exact $\infty$-categories and discuss the multiplicative properties of the Gabriel--Quillen embedding. For $E$ an Adams-type homotopy associative ring spectrum, this allows us to identify the symmetric monoidal $\infty$-category of $E$-based synthetic spectra with the presentable stable envelope of the exact $\infty$-category of compact spectra with finite projective $E$-homology. In addition, we show that algebraic K-theory, considered as a functor on exact $\infty$-categories, admits a unique delooping as a localising invariant.