On exact categories and their stable envelopes

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.

28 pages; many small improvements and expanded discussion of weight structures following referee reports; this version of the article has been accepted for publication, after peer review, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s00209-025-03904-6

References in corpus (7)

• Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
• A universal characterization of higher algebraic K-theory
• On the algebraic K-theory of higher categories
• Universality of multiplicative infinite loop space machines
• On exact infinity-categories and the Theorem of the Heart
• A counterexample to vanishing conjectures for negative $K$-theory
• The stable hull of an exact $\infty$-category