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.

- true
**?**- true
- false