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Texified, since problem was on front-page anyway

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edited May 8 2012 at 0:58

David White
5,050●1●20●27

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 y^{p$y^p$ is py^{p-1$py^{p-1} = 0 0$ by the Leibniz rule, so [y^p] $[y^p]$ 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.

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answered May 7 2010 at 19:17

Tyler Lawson
19k●3●47●93

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 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_∞ DGAs.)

Nothing goes wrong in characteristic zero.