How many model categories have the same weak equivalences?

There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model category M and construct a new model category structure on M with the same cofibrations, but with more weak equivalences and fewer fibrations. In these settings the identity adjoint equivalence often serves as a Quillen pair relating the Model categories.

One classic example is when the underlying category is a the category of functors from a (small) category C into a (nice) model category M. In that case, we have two naturally occuring model structures: the projective and the injective model structures. In some cases, when the source category C is a Reedy category, there is a third model structure on D=Fun(C,M) known as the Reedy model structure. All three of these model category structures are Quillen equivalent. In fact the weak equivalences are exactly the same. They are the level-wise weak equivalences.

In some sense which I'm trying to make precise, the injective and projective model structures on on opposite sides of the spectrum. The Reedy model structure is something like a mix of these two. In fact the model structures with a fixed set of weak equivalences should form a poset. For example we can just look at the set of cofibrations. Afterall the cofibrations and the weak equivalences determine the model structure if it exists. Then the model category structures with a fixed set of weak equivalences are ordered by inclusion (of sets of cofibrations). In the above example we have the ordering: $$Proj \subseteq Reedy \subseteq Inj$$

So this raises the question, what is known about different model structures on a category with a fixed set of weak equivalences? Is there always a maximal/minimal model structure? If not, are there some conditions which ensure its existence? Are there properties which characterize these model structures? Since the weak equivalences are always the same, then the identity functor should induce an isomorphism on homotopy categories. Thus if two such model structure are comparable (so that the identity adjoint equivalence is a Quillen pair) then they are Quillen equivalent. So if there is a minimal or maximal model structure, they all these model structures are Quillen equivalent via a zig-zag of Quillen equivalences. Is this the case?

• ams.org/mathscinet-getitem?mr=2203016 – Ben Wieland Jul 1 '10 at 5:59
• Seems like Tibor Beke's notes are probably the best you're going to do for an answer. Which is to say, the answer is not very bad for combinatorial model categories, but it can be quite messy without that hypothesis. One thing I didn't notice in Beke's notes was the notion of Left Determined Model Category, which might be a useful analogy (but one where you fix the cofibrations and vary the weak equivalences). Was there anything else you hoped to get when asking this question? If so, I'd be interested in thinking about it. If not, maybe you should accept Gabriel's answer. – David White Feb 26 '13 at 22:29