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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 otherwords (replaceing 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:

  1. 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?

  2. Is there an illuminating example for how the projective model structure can fail to be cartesian?

  3. (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 implies 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?

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One sufficient condition is that $D$ has coproducts (and of course that $\mathcal{M}$ is cartesian itself). Then the pushout product of generating projective cofibrations of $\mathcal{M}^D$ is a cofibration in $\mathcal{M}$ tensored with a representable copresheaf on $D$ so it is again a projective cofibration. This handles Segal's $\Gamma$ and the site of smooth manifolds (if we allow disconnected manifolds with components of varying dimensions). –  Karol Szumiło Mar 6 '13 at 11:01
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3 Answers 3

up vote 5 down vote accepted

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, nLab seems to say for $D=sSet^{op}$ that $M^D$ is cartesian closed, 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. If I find this I'll edit, but I'm less hopeful now than I was this morning of my ability to find it. If there was more time I'd just reconstruct it, but I'll be travelling for the next few days and probably won't have time.

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. It's Lemma 4.2.

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, the projective model structure with product $f\otimes g$ always satisfies the pushout product axiom. For the injective model structure you can get a monoidal model category via the pushout product $f \Box g$. This can be found in Hovey's book and actually a lot more can be said. Hovey has a preprint studying these structures in detail, but because it's not up online yet I don't want to steal his thunder by talking about it here.

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I feel very out-classed by the other answerers, but I'll make my edit anyway and hopefully make things a bit clearer in my answer –  David White Mar 6 '13 at 20:35
    
The preprint I mention in the last paragraph is now available at: arxiv.org/abs/1401.2850 –  David White Feb 3 at 15:36
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Hi Chris, here is just an answer to your "Main question". I hope it responds to what you had in mind. It is Lemma 1.2 in the current draft of "Enriched model categories and presheaf categories", by Bertrand Guillou and myself, which has been totally revised from the version on the archives (which also has this lemma, but I don't remember where). The revision will be posted soon. Context: the paper is all about the projective model structure in enriched categories in general; $\mathcal{V}$ is the enriching category and a $\mathcal{V}$-model category means what you would expect: $\mathcal{V}$-bicomplete and the evident behavior of the internal hom $\underline{M}(-,-)$ wrt cofibrations and fibrations. Tensors are denoted by $\odot$. No claim to originality, but we didn't know a reference.

Lemma. Let $\mathcal{M}$ be a $\mathcal{V}$-model category, let $M$ and $M'$ be cofibrant objects of $\mathcal{M}$, and let $N$ and $N'$ be fibrant objects of $\mathcal{M}$. If $\zeta\colon M\longrightarrow M'$ and $\xi\colon N\longrightarrow N'$ are weak equivalences in $\mathcal{M}$, then the induced maps $$ \zeta^*\colon \underline{\mathcal{M}}(M',N) \longrightarrow \underline{\mathcal{M}}(M,N)$$ and $$ \xi_*\colon \underline{\mathcal{M}}(M,N) \longrightarrow \underline{\mathcal{M}}(M,N')$$ are weak equivalences in $\mathcal{V}$.

Proof. We prove the result for $\xi_*$. The proof for $\zeta^*$ is dual. Consider the functor $\underline{\mathcal{M}}(M,-)$ from $\mathcal{M}$ to $\mathcal{V}$. By Ken Brown's lemma and our assumption that $N$ and $N'$ are fibrant, it suffices to prove that $\xi_*$ is a weak equivalence when $\xi$ is an acyclic fibration. If $V\longrightarrow W$ is a cofibration in $\mathcal{V}$, then $M\odot V \longrightarrow M\odot W$ is a cofibration in $\mathcal{M}$ since $M$ is cofibrant and $\mathcal{M}$ is a $\mathcal{V}$-model category. Therefore the adjunction that defines $\odot$ implies that if $\xi$ is an acyclic fibration in $\mathcal{M}$, then $\xi_*$ is an acyclic fibration in $\mathcal{V}$ and thus a weak equivalence in $\mathcal{V}$.

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Thank you Peter. The arxiv version of your paper seems full of useful results. I look forward to the update. However I don't think the results you mention apply to the question I intended to ask. Namely it seems you are talking about the $\mathcal{V}$-enriched internal hom, and I am after the $\mathcal{M}$-enriched internal hom. So in the limited setting I consider above, your results imply that the projective model structure on simplicial presheaves is a simplicial model category (this fact is stated in Charles' paper too). (continued) –  Chris Schommer-Pries Mar 6 '13 at 15:49
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However, I want to know whether the projective model structure is enriched over itself, i.e. whether $\mathcal{M}$ is a $\mathcal{M}$-model category? (Your definition of $\mathcal{M}$-model category supposes (an equivalent form of) the pushout-product axiom, so asking if $\mathcal{M}$ is cartesian is essentially equivalent to asking if it an $\mathcal{M}$-model category in your sense). I haven't read your entire paper, so maybe there is something in there addressing this? –  Chris Schommer-Pries Mar 6 '13 at 15:56
    
Chris: you're correct that $M$ is monoidal iff $M$ is an $M$-model category (another reference for a model category "over" another model category is Hovey's book). Unless I'm reading this wrong, in the special case where $V=M$, this answer seems to be recovering the part of your question where you mentioned that $A,X\mapsto Hom(A,X)$ sends weak equivalences to weak equivalences. If we're talking about $M^D$ as an $M$-model category could this result be used to answer your main question? –  David White Mar 6 '13 at 20:52
    
Right Chris, for me the cartesian context is foreign (since it is not a natural one in the applications that interest me most). "Enriched over itself" presumably means that you are thinking of your pre sheaf categories as cartesian monoidal. I was thinking of them as enriched over V, as you say, and in the examples there can be quite different relevant monoidal structures (even when V is simplicial sets) for which the pushout product axiom is not hard to check. The "equivalent form" you refer to involves tensors, not products. –  Peter May Mar 8 '13 at 4:30
    
And I should apologize for not reading your whole question carefully. –  Peter May Mar 8 '13 at 4:32
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Just a remark about making the projective model structure on simplicial presheaves (on a category $C$) cartesian.

In the projective model structure, representable objects are cofibrant, (and in some sense all cofibrant objects are built from them). So a helpful condition to impose might be:

  • finite products of representable functors are projective cofibrant.

This is clearly a necessary condition. Perhaps it is also sufficient?

One way to satisfy this is to assume that $C$ itself has finite products, so finite products of representables are representables. This sounds like the result of Sinan Yalin that D. White quotes.

Your $(0\to 1)$ example satisfies this property.

Another example which satisfies the condition is $C=$ discrete group $G$. The free functor is $G$ as a $G$-set, and $G\times G$ has a free $G$-action, so is cofibrant. (Though it is not itself a representable functor.) The projective model structure on $\mathrm{Psh}(G,\mathrm{sSet})$ is in fact cartesian.

(I should add that the fact that the injective model structure on simplicial presheaves is cartesian is not my observation: it is clear from Jardine's papers on the Joyal-Jardine model structures for simplicial (pre)sheaves, for instance.)

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Hi Charles, I like your idea about finite products of representables being cofibrant. Since finite products in C are the same as finite coproducts in $D = C^{op}$, this must be the same as the observation of D. White (quoting S. Yalin) and K. Szumilo. Of course the site of all manifolds of all dimensions has finite products, so I gather simplicial presheaves with the proj model str on this site should be cartesian. But if we fix the dimension we no longer have products. It seems weird that we would loose the cartesian property. This is probably a good test case for your conjectured criterion. –  Chris Schommer-Pries Mar 6 '13 at 21:36
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