An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's limits lie, especially in cases where that's not immediately obvious.
Quillen model categories are a classic such case. There are some easy rough intuitions—“something like topological spaces”, “somewhere one can talk about homotopy”, and so on—but various surprising examples show quite how crude those intuitions are, and persuade one that model categories cover a much wider range of situations than one might think at first.
However, I haven't seen any non-examples of model structures written up, or even discussed—that is, categories and classes of maps which one might think would be model structures, but which fail for subtle/surprising reasons. Presumably this is because, given the amount of work it typically takes to construct an interesting model structure, no-one wants to write (or read) three-quarters of that work without the payoff of an actual example at the end. Has anyone encountered any interesting non-examples of this sort?
Background on my motivations: I'm currently working with Batanin/Leinster style weak higher categories, and have a problem which seems amenable to model-theoretic techniques, so I'm trying to see if I can transfer/adapt/generalise the model structures defined by Cisinski et al, Lafont/Métayer/Worytkiewicz, etc. in this area. So I have some candidate (cofibrantly generated) classes of maps, and am trying to prove that they work; and there are lots of good examples around of how to prove that something is a model structure, but it would also be helpful to know what kinds of subtleties I should be looking out for that might make it fail to be.
Best Answer
Here is a classical example.
Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic $p$. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps $A \rightarrow B$ such that on the level of the underlying DGA, $B$ is a polynomial algebra over $A$ on a generator $x$ whose boundary is in $A$.
CDGA is complete and cocomplete, satisfies the $2$-out-of-$3$ axiom, the retract axiom, satisfies lifting, and a general map can be factored into a cofibration followed by an acyclic fibration by the small object argument.
However, you don't have factorizations into acyclic cofibrations followed by fibrations, because of the following.
Suppose $A \rightarrow B$ is a map of commutative DGAs which is a fibration in the above sense. Then for any element $[x]$ in the (co)homology of $B$ in even degree, the $p$-th power $[x]^p$ is in the image of the cohomology of $A$. In fact, pick any representing cycle $x \in B$ and choose a lift $y \in A$. Then the boundary of $y^p$ is $py^{p-1} = 0$ by the Leibniz rule, so $[y^p]$ is a lift of $[x]^p$ to the (co)homology of $A$.
(As a result, there are a lot of other "homotopical" constructions, such as homotopy pullbacks, that are forced to throw you out of the category of commutative DGAs into the category of $E_\infty$ DGAs.)
Nothing goes wrong in characteristic zero.