In his paper Homological algebra of homotopy algebras, Hinich defines a model structure on unbounded chain complexes where
- weak equivalences are the quasi-isomorphisms,
- fibrations are degree-wise surjections, and
- cofibrations are defined by the left lifting property.
Obviously, every object is fibrant. Now assume that we are working over a field $k$ instead of a general ring.
Question: Is it true that in this case every chain complex is also cofibrant?
I believe the answer to be yes, and I have an ugly argument that should prove it (even though I still have to check the details). Is there a sly/elegant way to show it?