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
2
-out-of-3
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
∈
B
and choose a lift ∈
y
∈
A
. Then the boundary of
y
p
is −1=0
p
y
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
∞
DGAs.)
Nothing goes wrong in characteristic zero.