Simplicial presheaves enriched, tensored and cotensored over simplicial sets correctly? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:17:23Z http://mathoverflow.net/feeds/question/87131 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87131/simplicial-presheaves-enriched-tensored-and-cotensored-over-simplicial-sets-corr Simplicial presheaves enriched, tensored and cotensored over simplicial sets correctly? dhagbert 2012-01-31T12:58:08Z 2012-07-10T03:03:06Z <p>Let <code>$\mathcal{C}$</code> be a category and <code>$\mathcal{P}=Functors(\mathcal{C}^{op},\mathcal{S})$</code> the category of simplicial presheaves, where <code>$\mathcal{S}=sSet$</code>. I want <code>$\mathcal{P}$</code> 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.)</p> <p>First, to any simplicial set <code>$K \in \mathcal{S}$</code>, I can associate a constant simplicial presheaf, also denoted <code>$K$</code>. Then I can define an action of the symmetrical monoidal category <code>$sSet$</code> on <code>$\mathcal{P}$</code> by taking the (categorical) product with <code>$K$</code> (which is computed levelwise). This, I think, is going to be the tensoring. </p> <p>Next, for any two <code>$F,G \in \mathcal{P}$</code>, we want mapping spaces <code>$Map_{\mathcal{P}}(F,G) \in \mathcal{S}$</code> so that the <code>$0$</code>-simplices are just the morphisms between <code>$F,G$</code> in <code>$\mathcal{P}$</code>, and so that this is compatible with the tensoring: <code>$Map_{\mathcal{P}}(K \times F,G) \simeq Map_{\mathcal{S}}(K, Map_{\mathcal{P}}(F,G))$</code>. In particular, setting $K=\Delta^{n}$, we see that we must have for the $n$-simplices <code>$Map_{\mathcal{P}}(F,G)_{n}=Hom_{\mathcal{P}}(\Delta^{n} \times F,G)$</code>.</p> <p>To the categorical product with constant simplicial presheaves and the above mapping spaces should make <code>$\mathcal{P}$</code> tensored and enriched over <code>`$\mathcal{S}$</code>.</p> <p>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 <code>$x \in \mathcal{C}$</code> is <code>$\mathcal{Hom}(F,G)(x)=Map_{\mathcal{P}}(F_{| \mathcal{C}/x},G_{| \mathcal{C}/x})$</code>, and the cotensoring $F^{K}$ should just be $\mathcal{Hom}_{\mathcal{P}}(K,G)$.</p> http://mathoverflow.net/questions/87131/simplicial-presheaves-enriched-tensored-and-cotensored-over-simplicial-sets-corr/101822#101822 Answer by BogdanG for Simplicial presheaves enriched, tensored and cotensored over simplicial sets correctly? BogdanG 2012-07-10T03:03:06Z 2012-07-10T03:03:06Z <p>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.</p> <p>First, $(\textbf{sSet}, \times, \ast)$ is a closed, symmetric monoidal category, where the internal-hom is $\textbf{Map}(F \times \Delta[-],G)$. </p> <p>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.</p> <p>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.</p> <p>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.</p> <p>For the model structures (projective and injective) on the category of simplicial presheaves you can have a look to <a href="http://dl.dropbox.com/u/2596527/MotivicHomotopyTheory.pdf" rel="nofollow">chapter 3 here</a>. The projective is very good since it is in fact monoidal,simplicial and proper, while the injective is only simplicial and proper.</p>