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 $f : X \rightarrow Y$ 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 $X \rightarrow A$ 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.)
Let me explain the motivation behind this question. I am trying to get a better picture of what category theory without the axiom of choice looks like.
My rough understanding is that in the absence of the axiom of choice we should use anafunctors in place of functors in some if not all contexts. An anafunctor from $C$ to $D$ is a span of functors $C \leftarrow E \rightarrow D$ in which the left leg $E \leftarrow C$ is a surjection on objects and fully faithful. The point is that even under these conditions $E \rightarrow C$ may not have a section. For instance, suppose $C$ is a category with all binary products. There may not be a product functor $- \times -: C \times C \rightarrow C$, because we cannot simply choose a distinguished product of each pair of objects. However, if we define $Prod(C)$ to be the category whose objects are diagrams of the shape $\bullet \leftarrow \bullet \rightarrow \bullet$ in $C$ which express the center object as the product of the outer two, there is a forgetful functor $Prod(C) \rightarrow C \times C$ remembering only the outer objects, and it is surjective on objects (because $C$ has binary products) and fully faithful. Furthermore there is another forgetful functor $Prod(C) \rightarrow C$ which remembers only the center object. Together these define a product anafunctor $C \times C \leftarrow Prod(C) \rightarrow C$.
"Classically" (:= under $AC$) the following paragraph holds: There is a model category structure on $Cat$ in which the cofibrations are functors which are injective on objects, fibrations are the functors with the right lifting property w.r.t. the inclusion of an object into the contractible groupoid on two objects, and weak equivalences are equivalences of categories. In particular the acyclic fibrations are the functors which are surjective on objects and fully faithful, exactly the functors we allow as left legs of anafunctors. Since every category is fibrant in this model structure, we can view an anafunctor from $C$ to $D$ as a representative of an element of $RHom(C, D)$, i.e., a functor from a cofibrant replacement for $C$ to $D$. Of course, every category is cofibrant too so for our cofibrant "replacement" we can just take $C$, and we learn that anafunctors from $C$ to $D$ are the same as functors when we consider both up to natural isomorphism (homotopy).
I would like to understand anafunctors as a kind of $RHom$ also without the axiom of choice. But I cannot use the same definition of the model category structure, because the lifting axioms require $AC$. I would like to keep the same acyclic fibrations, since they appear in the definition of anafunctor, and I would like every object to be fibrant. I cannot really imagine what a cofibrant replacement could look like in this model structure, but then I am not accustomed to working without $AC$.
My original question is related to this one via the functor which assigns to a set $S$ the codiscrete category on $S$, and it seems to contain the same kinds of difficulties.
Best Answer
Indeed, COSHEP (more traditionally called the "presentation axiom" by constructivists) does seem to be what you need in order to get a model structure on Set, or Cat. That's true in a lot of similar cases: cofibrant objects are always "projective" in some sense, and you can't expect to get many projective objects in a category constructed from sets unless you started with enough projective objects in Set itself. So I don't expect that ordinary model category theory will be much good for anything at all if you don't have at least COSHEP. But, thankfully, model categories are not the be-all and end-all of homotopy theory, so we can still formally invert the weak equivalences (functors that are fully faithful and essentially surjective) to obtain anafunctors as a derived hom.