What this boils down to is: if $\mathcal{K}$ is cofibrantly-generated model category which permits the small object argument and $\mathcal{D}$ is a small category, then when does $\mathcal{K}^\mathcal{D}$ permit the small object argument? This might be trivial depending on what exactly it means to "permit the small object argument".
Since this is really a question about Quillen's small object argument, allow me to frame it in a minimal way. Let $\mathcal{K}$ be a cocomplete category and let $I$ be a set of morphisms in $\mathcal K$. Recall that the small object argument says:
If $(\mathcal K,I)$ permits the small object argument, then $(\mathrm{cof}(I),I^\square)$ forms a weak factorization system on $\mathcal K$.
Here $\mathrm{cof}(I)$ is the closure of $I$ under coproducts, retracts, cobase change, and transfinite composition, and $I^\square$ is the class of morphisms which have the right lifting property with respect to the maps of $I$.
Depending on the author, "permitting the small object argument" can mean various things. Some authors use a strong sense which I am not so interested in:
- $(\mathcal{K},I)$ permits the small object argument if: Every object of $\mathcal{K}$ which is the domain of an arrow in $I$ is small.
(1) encompasses many important examples, including the weak factorization systems found in all combinatorial model categories. But (1) does not encompass the weak factorzation systems found in such important model cateogories as $\mathsf{Top}$ (in fact, the only small objects of $\mathsf{Top}$ are the discrete spaces), so it is sometimes necessary to use the more liberal sense which I am more interested in:
- $(\mathcal{K},I)$ permits the small object if: Every object of $\mathcal{K}$ which is the domain of an arrow in $I$ is $\mathrm{cof}(I)$-small.
Recall that an object $A$ of a category $\mathcal{K}$ is called small (or presentable) if there exists a regular cardinal $\lambda$ such that the covariant hom-functor $\mathcal{K}(A,-)$ preserves $\lambda$-filtered colimits. Here, if $S$ is a collection of morphisms in $\mathcal{K}$, I say that $A$ is $S$-small if $\mathcal{K}(A,-)$ preserves $\lambda$-filtered colimits over functors that land in $S$.
Now, if $(\mathcal{K},I)$ permits the small object argument and $\mathcal{D}$ is a small category, I'm interested in knowing
When does $(\mathcal{K}^\mathcal{D}, I^\mathcal{D})$ permit the small object argument?
Here $I^\mathcal{D} = \{ \mathcal{D}(D,-) \cdot i \mid D \in \mathcal{D}, i \in I\}$ where "$\cdot$" is a tensor, a.k. copower. Under sense (1), it's not hard to see that the answer is: always. But under sense (2), it's not so clear. I don't even see why the domains of $I^\mathcal{D}$ should be small with respect to the maps of $I^\mathcal{D}$ themselves, nevermind $\mathrm{cof}(I^\mathcal{D})$! (If this can be established, I think I can see that the class of maps with respect to which $A$ is $\lambda$-small is closed under retract and transfinite composition of length $< \lambda$, but I don't see why it should be closed under cobase change.) So if anyone can help me out with an argument, or at least a pointer to the literature, I'd appreciate it!
(The connection to the projective model structure is that if $(\mathcal{K},I)$ is a cofibrantly-generated model category permitting the small object argument and $\mathcal{D}$ is a small category, then I thought that $(\mathcal{K}^\mathcal{D},I^\mathcal{D})$ was supposed to form a cofibrantly generated model category. But the proof requires $(\mathcal{K}^\mathcal{D},I^\mathcal{D})$ to permit the small object argument. The nlab article on the projective model structure actually currently neglects to mention the hypothesis of permitting the small object argument entirely. The nlab's main reference is the 2nd appendix of Higher Topos Theory, where Lurie simply requires requires that $\mathcal{K}$ be combinatorial, so that (1) holds. Emily Riehl also uses sense (1) in her book. She mentions that Hovey uses sense (1) in his book, but I can't find a discussion of the projective model structure in Hovey. In fact, I was surprised at how difficult it was to find an account of the projective model structure! I also looked in Hirschhorn, Goerss-Jardine, and Quillen's original monograph.)
