Let $\mathcal{C}$ be a complete and cocomplete category, and assume I have a set $W$ of morphisms in $\mathcal{C}$ which is closed under retracts and satisfies the 2-out-of-3 property. Assume now that I have another set $F$ of morphisms, also closed under retracts. Is there a characterization of the properties that I need on $F$ so that it is the set of fibrations in a model structure on $\mathcal{C}$ with $W$ as the set of weak equivalences (with cofibrations defined by lifting property)?
I know that many will think of fibrantly generated model categories reading this question, but I would like $F$ to be the whole set of fibrations, and not just a set generating them (as morphisms with the lifting property with respect to the cofibrations).
Also, please specify what axioms you are using in the definition of model category. I am using the ones found in the paper On PL De Rham theory and rational homotopy type by Bousfield and Gugenheim, that are equivalent to the ones originally given by Quillen for closed categories, but I'm not averse to using slightly different axioms (e.g. requiring/implying functoriality of the factorizations of morphisms, as in Goerss-Jardine if I'm not mistaken).