It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
- cofibrations and weak equivalences
- fibrations and weak equivalences
The second statement follows immediately from the first by duality.
What about the following classes of maps/objects (A short argument would be very helpful)?
- cofibrations and fibrations
- cofibrant objects and weak equivalences
- cofibrant objects and fibrations
- cofibrant objects and fibrant objects
I think each of these classes determine the structure respectively. For the last one I suppose that one has to use framings but I cannot see how to do it.
Edit: Thank you all for the illuminative answers.