On exact categories and their stable envelopes

TL;DR

This paper establishes a deep connection between stable $ty$-categories and exact $ty$-categories through Klemenc's stable envelope, generalizes the Gillet-Waldhausen theorem, and explores the universal properties of algebraic K-theory.

Contribution

It introduces a new equivalence between stable $ty$-categories with bounded hearts and exact $ty$-categories, and extends key theorems to the setting of $ty$-categories.

Findings

Equivalence between stable $ty$-categories and exact $ty$-categories via stable envelope.

Generalization of the Gillet-Waldhausen theorem to connective algebraic K-theory.

Identification of a universal property of connective algebraic K-theory.

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.