This week, we learned that another example is the category of simplicial sets, and the class of weak equivalences the simplicial homotopy equivalences. All credit to Tom Goodwillie, Tim Campion, and Tyrone for figuring out why such a model structure cannot exist.

This week, we learned that another example is the category of simplicial sets, and the class of weak equivalences the simplicial homotopy equivalences. All credit to Tom Goodwillie, Tim Campion, and Tyrone for figuring out why such a model structure cannot exist.

I made this answer CW, since I'm really just advertising answers written by two other people.