Projective model structure on nonnegative chain complexes (${\mathbf{Ch}}_{\ge 0}(R)$)

The model category of nonnegative complexes of left modules has quasi-isomorphisms as model weak equivalences, positive-degree surjections as model fibrations, and degreewise injections with projective cokernel as model cofibrations. All objects are fibrant; cofibrant objects are degreewise projective.

The sphere chain complex inclusions into disk chain complexes, together with the separate degree-zero cell, generate the model cofibrations by cell attachment and retracts. Their right-injective class is exactly the acyclic fibration in the projective chain-complex model structure.

The right lifting property for each sphere-to-disk map fills a prescribed cycle and a compatible target element. The degree-zero generator supplies initial surjectivity; all other fillers force an acyclic kernel and the remaining degreewise surjections.

A map that is both a model fibration and a quasi-isomorphism. The homology isomorphism together with degree-one surjectivity forces surjectivity in degree zero. Equivalently it has the right lifting property against the generating cofibrations for nonnegative chain complexes.