On exact categories and their stable envelopes

We show that Klemenc’s stable envelope of exact ∞ ∞ -categories induces an equivalence between stable ∞ ∞ -categories with a bounded heart structure and weakly idempotent complete exact ∞ ∞ -categories. Moreover, we generalise the Gillet–Waldhausen theorem to the connective algebraic K-theory of exact ∞ ∞ -categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact ∞ ∞ -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.