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Let $\mathcal{C}$ be a category and $\mathcal{P}=Functors(\mathcal{C}^{op},\mathcal{S})$ the category of simplicial presheaves, where $\mathcal{S}=sSet$. I want $\mathcal{P}$ to be enriched, tensored, and cotensored over the category of simplicial sets $\mathcal{S}$ in the correct way. So I'll spell this out myself, and then the question will be if I got it right. (I'm pretty sure the answer is yes, but I'm uncertain. A reference where this is spelled out would be very welcome.)

First, to any simplicial set $K \in \mathcal{S}$, I can associate a constant simplicial presheaf, also denoted $K$. Then I can define an action of the symmetrical monoidal category $sSet$ on $\mathcal{P}$ by taking the (categorical) product with $K$ (which is computed levelwise). This, I think, is going to be the tensoring.

Next, for any two $F,G \in \mathcal{P}$, we want mapping spaces $Map_{\mathcal{P}}(F,G) \in \mathcal{S}$ so that the $0$-simplices are just the morphisms between $F,G$ in $\mathcal{P}$, and so that this is compatible with the tensoring: $Map_{\mathcal{P}}(K \times F,G) \simeq Map_{\mathcal{S}}(K, Map_{\mathcal{P}}(F,G))$. In particular, setting $K=\Delta^{n}$, we see that we must have for the $n$-simplices $Map_{\mathcal{P}}(F,G)_{n}=Hom_{\mathcal{P}}(\Delta^{n} \times F,G)$.

To the categorical product with constant simplicial presheaves and the above mapping spaces should make $\mathcal{P}$ tensored and enriched over $\mathcal{S}$.

Finally, we want $\mathcal{P}$ to be 'cotensored' or 'powered'. In fact, I think more is true. $\mathcal{P}$ should have an internal Hom whose value at $x \in \mathcal{C}$ is $\mathcal{Hom}(F,G)(x)=Map_{\mathcal{P}}(F_{| \mathcal{C}/x},G_{| \mathcal{C}/x})$, and the cotensoring $F^{K}$ should just be $\mathcal{Hom}_{\mathcal{P}}(K,G)$.

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I didn't read every line of your post, so I don't want to comment on correctness. However, Hovey has written about these things, as has his student Gillespie. Might be worth googling. Of course, on the homotopy level you should have no problem being tensored, cotensored, and enriched over $SSet$ (see e.g. Spectra and Symmetric Spectra in General model categories). But to get this on the model category level will also require the Pushout Product axiom, which I don't see appearing in your write-up so far. –  David White Jan 31 '12 at 18:09

1 Answer 1

Hi, yes you are right. I don't know if you're still interested, but since I took some time to understand what happens here, I might share it.

First, $(\textbf{sSet}, \times, \ast)$ is a closed, symmetric monoidal category, where the internal-hom is $\textbf{Map}(F \times \Delta[-],G)$.

For any small category $\mathcal{C}$, the category of simplicial presheaves $[\mathcal{C}^{\text{op}}, \textbf{sSet}]$ inherits object-wise the monoidal structure, that is, $(F \times G)(c) := F(c) \times G(c) \in \textbf{sSet}$ and the unit is just the object-wise unit. This is trivially symmetric, and in fact also closed with the object-wise internal-hom $(F^G)(c) := F(c)^{G(c)} = \textbf{Map}(G(c), F(c)) \in \textbf{sSet}$. No tricks, everything is object-wise.

Now, there is a fully faithful embedding $\textbf{sSet} \hookrightarrow [\mathcal{C}^{\text{op}}, \textbf{sSet}]$ sending a simplicial set to the constant (on objects) diagram. Therefore, there is a now a notion of tensor product between a simplicial set $X_{\bullet}$ and a simplicial presheaf $F$, by doing the product in the category of simplicial presheaves after the above embedding, i.e., $(X_{\bullet} \otimes F)(c) := X_{\bullet} \times F(c) \in \textbf{sSet}$ and as you said, this is the tensor. The enriched-hom is now just $\textbf{Map}(F \otimes \Delta[-], G)$, and the cotensor similarly $(F^{X_{\bullet}})(c) := \textbf{Map}(F(c), X_{\bullet})$. This gives the $\textbf{sSet}$-enrichment of the category of simplicial presheaves which is tensored and cotensored.

So there are two structures on the category of simplicial presheaves : it is first symmetric closed monoidal and so it can be enriched over itself with a simplicial presheaf as internal-hom, and it also is enriched over simplicial sets which is just a full subcategory and gives a simplicial mapping space.

For the model structures (projective and injective) on the category of simplicial presheaves you can have a look to chapter 3 here. The projective is very good since it is in fact monoidal,simplicial and proper, while the injective is only simplicial and proper.

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