On exact categories and their stable envelopes

TL;DR

The paper develops a comprehensive framework linking exact $\infty$-categories, stable envelopes, and $K$-theory. It shows that the stable envelope of an exact $\infty$-category carries a bounded heart whose heart is the weak idempotent completion, and it establishes an adjunction between exact $\infty$-categories and bounded-heart stable categories, unifying exact and stable perspectives. It generalises the Gabriel–Quillen embedding and Keller's criterion to the higher setting, proving that $K$-theory is invariant under stabilization and enjoys a universal, additive characterization via a universal property of $K$-theory on exact categories. These results also clarify how semi-orthogonal decompositions and weight/heart structures behave under stabilization and have implications for noncommutative motives. Overall, the work provides a robust, universal description of connective $K$-theory in the exact-category context and bridges foundational higher-categorical constructions with classical $K$-theory principles.

Abstract

We show that Klemenc's stable envelope of exact $\infty$-categories induces an equivalence between stable $\infty$-categories with a bounded heart structure and weakly idempotent complete exact $\infty$-categories. Moreover, we generalise the Gillet-Waldhausen theorem to the connective algebraic K-theory of exact $\infty$-categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact $\infty$-categories. A key tool is a generalisation of a theorem due to Keller which provides a sufficient condition for an exact functor to induce a fully faithful functor on stable envelopes.

Theorems & Definitions (89)

• Definition 1.1
• Theorem 1.2: Keller's criterion
• Proposition 1.3
• Theorem 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8: Gillet--Waldhausen
• Theorem 1.9
• Definition 2.1