# Fitting desired weak equivalences and cofibrations into a model category

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

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.