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Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and $\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are the weak equivalences and the cofibrations of a model category structure.

I can certainly write down what the fibrations $\mathcal{F}$ would have to be, and I'm wondering if there are any theorems to provide shortcuts in verifying that the classes $\mathcal{W}$, $\mathcal{C}$ and $\mathcal{F}$ actually satisfy all the conditions of a model category.

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Sure, e.g. Smith's recognition theorem for combinatorial model categories ncatlab.org/nlab/show/combinatorial+model+category#SmithTheorem –  Fernando Muro Jan 31 '12 at 16:00
You should look at this old thread, on a very similar question: mathoverflow.net/questions/84086. In particular, I give an answer there which goes into more detail on Gillespie's work than Tim's answer below. Also, Charles Rezk's answer on that thread is very good. –  David White Jan 31 '12 at 17:49

2 Answers 2

There was an important paper by K. S. Brown:“Abstract homotopy theory and generalized sheaf cohomology” , Trans. Amer. Math. Soc 186 (1973), p. 419 – 458. That gave a set of axioms for a fibration category. Baues (and I'm surprised that Fernando did not mention this) in Algebraic Homotopy gives a 'cofibration category' and similar structures are available in Kamps and Porter's textbook. These axiomatise the fibrations or the cofibrations in a neat way. The sticking point is often whether the limits and colimits needed are there.

You give no details about your category but if it is additive or better exact then look at J. Gillespie, Journal of Pure and Applied Algebra, 215 (2011) 2892–2902.

With more information a fuller answer can be found, but that will do for the moment!!!

Of course, I should also ask why do you think you need a full model category structure. You can do a lot even without it, although it is nice to have when it is available.

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I think that Jeff Smith's work is unpublished, but a general machine (presumably based on it) for constructing model structures from cofibrations and weak equivalences is in Lurie's "Higher Topos Theory" (see for instance A.2.6.13). Namely, one has to consider a class of cofibrations and weak equivalences on a nice (presentable) category, where the cofibrations have a generating set and the weak equivalences satisfy an accessibility condition: then, almost for free, you get a model structure with those cofibrations and weak equivalences. The thing to check is that a map with the right lifting property with respect to all the cofibrations is a weak equivalence.

Lurie applies this criterion repeatedly in his book, e.g. in constructing the co(ntra)variant, Joyal, and (co)Cartesian model structures. Unfortunately the procedure produces a set generating acyclic cofibrations by more or less choosing all "small" acyclic cofibrations, and as a result of this non-constructive procedure it seems rather difficult to determine what the fibrations are (at least between non-fibrant objects); the case of the usual model structure on simplicial sets seems to be one of the few cases where this has been done.

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