Projective cofibration structure on bounded-below complexes

Proof. The 2-out-of-3 property is clear. The stated cofibrations contain the isomorphisms and are closed under composition: the cokernel of a composite is a degreewise split extension of the two projective cokernels. They are closed under pushout, since a pushout is again degreewise split with the same cokernel. This also proves that pushouts of trivial cofibrations are weak equivalences, using Corollary 6.1.18.

It remains to verify the factorization axiom. Let \(f\colon P_{\bullet }\to C_{\bullet }\) be a chain map with \(P_{\bullet }\) cofibrant, i.e. bounded below projective. Choose a degreewise epimorphic projective resolution \(q\colon Q_{\bullet }\xrightarrow {\sim }C_{\bullet }\) using Proposition 6.6.13. Since \(P_{\bullet }\) is bounded below projective and \(q\) is a degreewise epimorphism with acyclic kernel, Lemma 6.6.14 provides a chain map \(\widetilde f\colon P_{\bullet }\to Q_{\bullet }\) with \(q\widetilde f=f\). Now consider the mapping-cylinder factorization of \(\widetilde f\) from the proof of Proposition 6.1.27: \[ P_{\bullet }\lhook \joinrel \longrightarrow \Cyl (\widetilde f)_{\bullet }\xrightarrow {\sim }Q_{\bullet }. \] The first map is degreewise the inclusion \(P_n \hookrightarrow P_n \oplus P_{n-1}\oplus Q_n\), hence a degreewise split monomorphism whose cokernel is degreewise \(P_{n-1}\oplus Q_n\) and thus bounded below projective; so it is a cofibration. The second map is a chain homotopy equivalence, and composing it with \(q\) therefore exhibits \(f\) as a cofibration followed by a quasi-isomorphism.

Finally, \(0\to P_{\bullet }\) is a cofibration precisely when \(P_{\bullet }\) is bounded below projective. □