There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are the maps f : X → Y such that X and Y are both empty or both nonempty.

In order for the lifting axioms to hold we need the axiom of choice.  Suppose we want to avoid the axiom of choice.  One option seems to be to replace "epimorphism" with "map which has a section" everywhere.  Can we instead leave the definition of fibration unchanged and change the definition of cofibration?

Note that if A is a cofibrant set in this hypothetical model structure then any surjection X → A has a section.  So it would be necessary that every set admits a surjection from a set A of this type, which seems rather implausible to me.  Perhaps the notion of model category needs to be modified in a setting without the axiom of choice.

(Apologies if this question turns out to be meaningless or trivial; I have not thought about it much nor do I often try to avoid using the axiom of choice.)