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?