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2 Texified, since problem was on front-page anyway

Here is a classical example.

Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic p. $p$. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps $A B \rightarrow B$ such that on the level of the underlying DGA, B $B$ is a polynomial algebra over A $A$ on a generator x $x$ whose boundary is in A.$A$.

CDGA is complete and cocomplete, satisfies the 2-out-of-3 $2$-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 $A B \rightarrow B$ is a map of commutative DGAs which is a fibration in the above sense. Then for any element [x] $[x]$ in the (co)homology of B $B$ in even degree, the p'th $p$-th power [x]p$[x]^p$ is in the image of the cohomology of A. $A$. In fact, pick any representing cycle $x \in B B$ and choose a lift $y to A\in A$. Then the boundary of yp$y^p$ is pyp-1$py^{p-1} = 0 0$ by the Leibniz rule, so [yp] $[y^p]$ is a lift of [x]p$[x]^p$ to the (co)homology of A.$A$.

(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$E_\infty$ DGAs.)

Nothing goes wrong in characteristic zero.

1

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 → 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 → 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 to A. Then the boundary of yp is pyp-1 = 0 by the Leibniz rule, so [yp] 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 DGAs.)

Nothing goes wrong in characteristic zero.