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](https://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories) 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?**




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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.



Non-examples of model structures, that fail for subtle/surprising reasons?
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.


