Question

$ \newcommand{\calc}{\mathcal C} \newcommand{\calf}{\mathcal F} \newcommand{\calw}{\mathcal W} \newcommand{\calm}{\mathcal M} \newcommand{\caln}{\mathcal N} \newcommand{\colim}{\operatorname{colim}} $

Here's something unusual that I noticed, but I don't know a single situation where the hypotheses apply. If you have any idea if there are any such situations (even silly ones) or not, let me know. And thanks to A. Berglund for listening, and D. Yau for pointing out why the example that I thought I had doesn't work (subcategory wasn't full).

Update February 24, 2015 Shortly after the original post, I. Patchkoria pointed out to me that the inclusion from Kelley Spaces to Topological Spaces satisfies the conditions of this theorem (see Hovey 2.4.22—2.4.23). In fact, Hovey uses an argument quite similar to the one below to establish a model structure on Kelley spaces.
With some additional work, M. Robertson, D. Yau, and I were able use similar arguments to prove the theorem that I had originally hoped to be a corollary to the one on this page. Our result appears in the preprint A simplicial model for infinity properads.

Update June 10, 2016 Emily Riehl helpfully pointed out that the inclusion i should preserve transfinite compositions.

Update January 1, 2017 A version of this appeared as Lemma 8.8 in the 2002 paper Model structures on the category of ex-spaces by Michele Intermont and Mark Johnson.

Theorem Suppose that $\calm$ is a cofibrantly generated model category with classes of maps $(\calw, \calf, \calc)$, $i : \caln \hookrightarrow \calm$ is a full subcategory inclusion, and

$\caln$ has all limits
$\caln$ has all colimits and $i$ preserves pushouts and sequential colimits
the generating (acyclic) cofibrations of $\calm$ are $iI$ and $iJ$, respectively, for some $I,J \subset \caln$.

Then $\caln$ is a cofibrantly generated model category with class of weak equivalences $\calw' = \calw \cap \caln$ and fibrations $\calf' = \calf \cap \caln$.

Proof: Let $K\in \{ I, J\}$. The key points to check are the following:

$K\text{-inj} = iK\text{-inj} \cap \caln \label{a inj}$
$K\text{-cell} \subset iK\text{-cell} \cap \caln \label{a cell}$
$iK\text{-cof} \cap \caln \subset K\text{-cof}. \label{a cof}$

To see (1), we use the fact that $\caln$ is a full subcategory of $\calm$. If $f\in \caln$ has the rlp in $\calm$ with respect to all elements of $iK$, then by fullness it has the rlp in $\caln$ with respect to all elements of $K$. The reverse implication is trivial. For (2), note that applying $i$ to a pushout of a map in $K$ gives a pushout of a map in $iK$, since $i$ preserves pushouts. Finally, the proof of (3) is a variation of that on (1).

Now that we have (1-3), we apply a recognition theorem for cofibrantly generated model categories [1, 2.1.19].

(1) It is clear that $\calw'$ satisfies 2-of-3 and is closed under retracts.
(2-3) we already know that the domains of $K \in \{I,J\}$ are small relative to $iK\text{-cell}$, hence must be small relative to the smaller class $K\text{-cell}$ (2).
(4) We have \[ \begin{aligned} J\text{-cell} &\overset{(2)}\subset iJ\text{-cell} \cap \caln \\ &\subset (\calw \cap iI\text{-cof}) \cap \caln \\ &= \calw' \cap (iI\text{-cof} \cap \caln) \\ &\overset{(3)}\subset \calw' \cap I\text{-cof}. \end{aligned} \]
(5-6) We have \[ I\text{-inj} \overset{(1)}= iI\text{-inj} \cap \caln = iJ\text{-inj} \cap \calw \cap \caln \overset{(1)}= J\text{-inj} \cap \calw' \]

Thus $\caln$ is a cofibrantly generated model category with $\calw' = \calw \cap \caln$ as the subcategory of weak equivalences.

The characterization of fibrations as $\calf' = \calf \cap \caln$ is just (1) for $K=J$.

$\square$

^{1. Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.}

January 2, 2015