Return to Answer

EDIT (3/8/14): There appears to be a minor error in the Spitzweck paper, so I've struck out that text above. The proof of Corollary 8 says to look at Theorem 5. But Theorem 5 requires the operad to be cofibrant in the model category of sequences, i.e. to be projectively cofibrant (nowadays this would be called $\Sigma$-cofibrancy for the operad). Unfortunately, $Com$ does not satisfy this in general, so I don't believe Corollary 8 any more. In particular, I recently proved that if Corollary 8 were true then in fact CDGA($\mathbb{F}_p)$ would be a (full) model category. Indeed, in any combinatorial situation you can get from a semi-model category to a model category, by Jeff Smith's recognition theorem. Spitzweck's Corollary 8 should have been a statement about the operad $E_\infty$, and then the result is related to the Mandell example mentioned above. Note that $Com$ satisfies the conditions of Theorem 5 in $Ch(k)$ if $k$ is a field of characteristic zero, because in this setting all symmetric sequences are projectively cofibrant (I learned this from recent papers of John Harper). For non-zero characteristic it should be easy to construct examples which show that $Com$ is not projectively cofibrant as a symmetric sequence.

EDIT (3/9/14): It appears I made a mistake in my earlier version of this answer (and in a recent edit), due to not fully understanding the notation in Markus Spitzweck's thesis. Corollary 8 which I discuss above in the struck out text is not actually about strictly commutative monoids and thus has nothing to do with Tyler's answer. Spitzweck uses the notation $Comm_C$ to denote algebras over a particular $E_\infty$ operad, and the $\Sigma$-cofibrancy of that operad assures us (by his Theorem 5) that algebras over the opeard have a semi-model structure. $Com$ is not $\Sigma$-cofibrant in general, and in positive characteristic, $CDGA(k)$ is not a semi-model category (I recently proved that if it were then in fact CDGA($k$) would be a (full) model category). Note that $Com$ is $\Sigma$-cofibrant in $Ch(k)$ if $k$ is a field of characteristic zero, because in this setting all symmetric sequences are projectively cofibrant (I learned this from recent papers of John Harper). For non-zero characteristic it should be easy to construct examples which show that $Com$ is not $\Sigma$-cofibrant.

The best reference for $J$-semi model categories is Markus Spitzweck's Operads, Algebras and Modules in General Model Categories. This defines them more generally than Hovey and develops a theory which mimics that in Hovey's book on Model Categories. Corollary 8 of Spitzweck shows that Tyler's example is a $J$-semi model category which is not a model category. It proves in particular that if $M$ is a cofibrantly generated left proper model category with domains of the generating cofibrations cofibrant and cofibrant unit, then commutative monoids in $M$ form a $J$-semi model category. This paper also shows that if $O$ is an operad which is cofibrant in the projective model structure on collections, then algebras over $O$ and modules over a cofibrant $O$-algebra $A$ have $J$-semi model structures. Getting actual model structures rather than $J$-semi model structures requires more hypotheses on the underlying category $M$, and is part of my current thesis work.

This is Theorem 4.3.1. Theorem 4.4.2 gives a hypothesis under which this almost model structure becomes a model structure. You need to know that given a chain of $h$-acyclic $h$-cofibrations $j_n:Z_n\to Z_{n+1}$ and given a compatible system $q_n:Z_n\to Y$ which gives $q:$colim $Z_n \to Y$ then colim $Nq_n \to Nq$ is an isomorphism in $C$. So the failure of these structures on $C$ to form a model structure has to do with the failure of this axiom, which I imagine is a bit surprising to those who don't work with model categories a lot.

The best reference for $J$-semi model categories is Markus Spitzweck's Operads, Algebras and Modules in General Model Categories. This defines them more generally than Hovey and develops a theory which mimics that in Hovey's book on Model Categories. Corollary 8 of Spitzweck shows that Tyler's example is a $J$-semi model category which is not a model category. It proves in particular that if $M$ is a cofibrantly generated left proper model category with domains of the generating cofibrations cofibrant and cofibrant unit, then commutative monoids in $M$ form a $J$-semi model category. This paper also shows that if $O$ is an operad which is cofibrant in the projective model structure on collections, then algebras over $O$ and modules over a cofibrant $O$-algebra $A$ have $J$-semi model structures. Getting actual model structures rather than $J$-semi model structures requires more hypotheses on the underlying category $M$, and is part of my current thesis work.

This is Theorem 4.3.1. Theorem 4.4.2 gives a hypothesis under which this almost model structure becomes a model structure. You need to know that given a chain of $h$-acyclic $h$-cofibrations $j_n:Z_n\to Z_{n+1}$ and given a compatible system $q_n:Z_n\to Y$ which gives $q:$colim $Z_n \to Y$ then colim $Nq_n \to Nq$ is an isomorphism in $C$. So the failure of these structures on $C$ to form a model structure has to do with the failure of this axiom, which I imagine is a bit surprising to those who don't work with model categories a lot.

EDIT (3/8/14): There appears to be a minor error in the Spitzweck paper, so I've struck out that text above. The proof of Corollary 8 says to look at Theorem 5. But Theorem 5 requires the operad to be cofibrant in the model category of sequences, i.e. to be projectively cofibrant (nowadays this would be called $\Sigma$-cofibrancy for the operad). Unfortunately, $Com$ does not satisfy this in general, so I don't believe Corollary 8 any more. In particular, I recently proved that if Corollary 8 were true then in fact CDGA($\mathbb{F}_p)$ would be a (full) model category. Indeed, in any combinatorial situation you can get from a semi-model category to a model category, by Jeff Smith's recognition theorem. Spitzweck's Corollary 8 should have been a statement about the operad $E_\infty$, and then the result is related to the Mandell example mentioned above. Note that $Com$ satisfies the conditions of Theorem 5 in $Ch(k)$ if $k$ is a field of characteristic zero, because in this setting all symmetric sequences are projectively cofibrant (I learned this from recent papers of John Harper). For non-zero characteristic it should be easy to construct examples which show that $Com$ is not projectively cofibrant as a symmetric sequence.

EDIT #2: In Chapter 4 of May and Sigurdsson's Parameterized Homotopy Theory another type of "almost model category" comes up. In particular, given a topological model category $C$ you can try to define an $h$-type model structure analogous to what is done for spaces. So a map is an $h$-fibration if it has CHP (i.e. the RLP with respect to $X\to Cyl(X)$ for all $X$), and is an $h$-cofibration if it has HEP (i.e. the LLP with respect to $Cocyl(X)\to X$ for all $X$). Say that $f$ is a strong $h$-fibration if it has the relative CHP, and a strong $h$-cofibration if it has the relative HEP. Then using $h$-fibrations and strong $h$-cofibrations OR using $h$-cofibrations and strong $h$-fibrations give two new structures which satisfy all the axioms for a model category except that the factorization axiom becomes

"Any map $f: X \to Y$ factors as $X \to Mf \to Y$ where the first map is a strong $h$-cofibration and the second has a section which is an $h$-acyclic strong $h$-cofibration, and $f$ factors as $X\to Nf \to Y$ where the first map is a strong $h$-fibration and the second has a section which is an $h$-acyclic strong $h$-fibration."

This is Theorem 4.3.1. Theorem 4.4.2 gives a hypothesis under which this almost model structure becomes a model structure. You need to know that given a chain of $h$-acyclic $h$-cofibrations $j_n:Z_n\to Z_{n+1}$ and given a compatible system $q_n:Z_n\to Y$ which gives $q:$colim $Z_n \to Y$ then colim $Nq_n \to Nq$ is an isomorphism in $C$. So the failure of these structures on $C$ to form a model structure has to do with the failure of this axiom, which I imagine is a bit surprising to those who don't work with model categories a lot.