The homotopy ∞-category of chain complexes is stable

Proof. The proof is the split analogue of the proofs of Proposition 6.1.27, Theorem 6.1.30. Equip \(\Ch (\Aa )\) with chain homotopy equivalences as weak equivalences and degreewise split monomorphisms as cofibrations. The mapping-cylinder factorization shows that every map factors as a cofibration followed by a weak equivalence. The pushout axiom follows from the criterion that a degreewise split monomorphism is a chain homotopy equivalence precisely when its cokernel is contractible. The dual argument uses degreewise split epimorphisms, so the localization \(\Kk (\Aa )\) admits finite limits and colimits.

For a chain complex \(C_{\bullet }\), the degreewise split short exact sequence \[ 0 \to C_{\bullet } \to \Cyl (C_{\bullet }\to 0) \to C[1]_{\bullet } \to 0 \] is sent to a cofiber sequence in \(\Kk (\Aa )\). Its middle term is contractible, so it identifies suspension with the shift \([1]\). Thus suspension is an equivalence, with inverse induced by \([-1]\), and Theorem 4.2.2 implies that \(\Kk (\Aa )\) is stable. □