I encountered an issue today that I can't resolve to myself:
Consider the projective model structure on chain complexes over a ring R (Ch(R)), bounded below if you like.
Projectives in Ch(R) are split-exact complexes which are level-wise projective (so in particular are contractible). Cofibrations in Ch(R) are inclusions with projective cokernel, and so every cofibrant object is projective (since the initial object is the zero complex). In particular every cofibrant object is contractible, and so since every object of Ch(R) has a cofibrant replacement, every chain complex has trivial homology????
This obviously can't be right, and I'm misunderstanding something, and any help to figure out what that is would be much appreciated.
Addition: After thinking about it for a little bit longer I think that the word "projective" in the definition of a cofibration in the projective model structure must mean "level-wise projective", as this resolves the question, and there is a lot of literature on constructing "almost projective" resolutions in derived categories, of which a level-wise projective resolution is an example. I think the reason for their use is that to ask for a (categorically) projective resolution in derived categories is usually too strong a requirement to be able to do anything with it, (as illustrated by the contractibility of projective objects in Ch(R)). I hope this paragraph is correct, but comments/answers/corrections still very welcome.