If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the levelwise weak equivalences.
- The projective model structure, which is characterized by the fibrations being the levelwise fibrations, and
- The injective model structure, which is characterized by the cofibrations being the levelwise cofibrations.
Here "nice" means cofibrantly generated in the projective case and combinatorial in the injective case. Now I am interested in the case that $M = sSet$, the category of simplicial sets with the usual Kan-Quillen model structure. In other words (replacing D by its opposite) I am interested in the projective and injective model structures on simplicial presheaves.
Let us say that a model category is a cartesian model category if it is cartesian closed and it satisfies the pushout-product axiom. This is the same as saying the product is a Quillen adunction of two variables. In particular it implies that if $A$ is cofibrant and $X$ is fibrant, then the functors:
$$ Hom(A, -) $$
$$ Hom(-, X) $$
are part of Quillen adjunctions. (Here "Hom" is the inner hom). In particular the assignment $$ A,X \mapsto Hom(A,X)$$
sends weak equivalences to weak equivalences, provided the As are cofibrant and the Xs are fibrant.
A catchy way to summarize this last observation is to say that the derived functor of internal hom is homotopically meaningful.
Now the category of simplicial presheaves is cartesian closed, i.e. it has products and an internal hom. In section 2 of Rezk's paper "A Cartesian presentation of $(\infty,n)$-categories" (arXiv:0901.3602), he reviews these model categories and states that the injective model structure is always cartesian.
So this raises some questions about the projective model structure:
When is the projective model structure a cartesian model structure? There are examples where the injective and projective model structure agree (e.g. D = pt), so presumably there are less severe conditions one can impose on D to ensure this happens?
Is there an illuminating example for how the projective model structure can fail to be cartesian?
(Main question) Setting aside the question of cartesian-ness, we can also ask about whether the internal hom is invariant under weak equivalences, always assuming the source is cofibrant and the target is fibrant. This is the question I am most interested in. It is of course implied by cartesian-ness of the projective model structure, but a priori seems weaker.
I tried to construct a counter example to 3 by looking at the case $D = (0 \to 1)$, the free-walking arrow. However in this case I was thwarted by the fact that simplical sets is a right proper model category. In this specific easy case (even though the injective and projective structures differ) the internal hom is invariant.
There are two classes of D which interest me the most. First there are combinatorial sorts of categories which show up often in the theory of higher categories. I am thinking now of things like $D = \Delta, \Theta_n$ or Segal's $\Gamma$. The other case I am interested in is when $D$ is something like the site of smooth manifolds. The general setting might be out of reach, but hopefully something can be said in these cases.
There are also a variety of related bonus questions:
What happens if we localize our model category? Does the invariance of internal hom persist? Does cartesian-ness?.
What can we say when simplicial sets is replaced with another nice cartesian model category? What if this model category is right proper?
Best Answer
I got interested in a similar issue last summer, namely: "When does passage to the diagram category preserve the pushout product axiom?" I ended up finding a paper on arXiv by Sinan Yalin called "Classifying Spaces and module spaces of algebras over a prop" which gives conditions on $M$ and $D$ so that $M^D$ satisfies the pushout product axiom. What's needed is that $D$ has finite coproducts (and of course that $M$ has the pushout product axiom). So that answers the monoidal model category part of (1).
To determine when $M^D$ is cartesian is a purely category theory question. I imagine this has been studied classically, e.g. in chapter 8 of Awodey's Category Theory. Also, Lemma 3 at nLab seems to say for $M=sSet$ that $M^D$ is cartesian closed (for sites $D$ with finite products), so your example of interest is taken care of. I'd love to see a characterization of when $M^D$ is cartesian closed. That would finish the answer of (1) and therefore (3).
For (2), I'm fairly certain that at one point over the summer I came up with an example showing that without the hypothesis regarding finite coproducts $M^D$ can fail to be a monoidal model category.
I was also curious about the injective and Reedy model structures. It's easy to see that passage to the injective model structure always preserves the pushout product axiom. Regarding the second part of your question (1), one such condition is that $D$ is Reedy (in which case the projective, injective, and Reedy structures all match). Barwick's paper "On Reedy Model Categories" gives conditions under which passage to $M^D$ with the Reedy model structure preserves the pushout product axiom. Corollary 4.17 discusses the case where the monoidal product is given by Day convolution. Theorem 4.18 discusses the case where it's given object-wise using the monoidal structure on $M$. This note of Barwick's is contained in his paper "On Left and Right Model Categories and Left and Right Bousfield Localizations" if you want a published reference.
For your question about localization, I can help a little bit. First, because you have the same objects and morphisms, localization will preserve the property of being cartesian closed. In my thesis, one thing I study is when Bousfield localization preserves the pushout product axiom. I work in the setting where we're cofibrantly generated and the domains of the generating maps are cofibrant. In that case, the correct hypothesis to conclude that localization (at a set of maps $T$) preserves the pushout product axiom is: "$T\otimes id_K$ is a $T$-local equivalence as $K$ runs over all (co)domains of the generating maps." The unit axiom is also easy. So this answers your bonus question. For me, right properness didn't come into it, but I did need left properness. You probably already need this, though, to do Bousfield localization at all. See my recent MO question if you want to try localization without left properness.
The case with $sSet$ is very nice, so probably you can get away with something simpler. Others have thought of questions like this before, but perhaps not in this level of generality. For instance, Barnes has a paper where he does this for stable model categories. So maybe there's a paper that does it for $sSet$ and provides a hypothesis which is easier to check.
Also, your example of $D$ being $\bullet \to \bullet$ turns out to behave very nicely. For instance, Yalin's result shows that the projective model structure with monoidal structure given by $f \otimes g:A\otimes X\to B\otimes Y$ satisfies the pushout product axiom as soon as $M$ does. Additionally, the projective model structure with monoidal structure given by the pushout product $f\Box g$ satisfies the pushout product axiom. For the injective model structure you can get a monoidal model category via the product $f \otimes g$. A reference for the last two statements is Hovey's paper Smith ideals of structured ring spectra and actually a lot more can be said (e.g. you can get the monoid axiom on these diagram categories).