I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial model category. So it would be convenient if I could have a simplicial model category model for the functor category.


I imagine doing something like the following (sketch):


1) find a model category which models the (infinity,1) category: $\textrm{Fun}(N^{hc}\_{\bullet}(M^{cf}),\textrm{SSet})$. I'll call such a model category $\textrm{Fun}(M,\textrm{SSet})$


2) use Bousfield localization on the collection of morphisms S = {for each functor F, the comparison maps F(lim d) ---> lim Fd } in $\textrm{Fun}(M,\textrm{SSet})$ to get a model category structure which models the category of homotopy limit preserving functors.


So my questions are 



> 
> Question 1: Given a simplicial model category $M$, what model category models the functors from $M$ to Spaces?
> 
> 
> Question 2: Given a simplicial model category $M$, what model category models the homotopy-limit-preserving functors from $M$ to Spaces?
> 
> 
> 


Answers to my question don't need to address my sketch, but I am curious about whether that will work. References would also be appreciated. 



Model for the (infinity,1)-category of (homotopy-)limit preserving functors
Suppose that the dual $M^{\mathrm{op}}$ of your original simplicial model category $M$ is combinatorial, so that its associated $\infty$-category $\mathcal{M}^{\mathrm{op}}$ is presentable. Then what you are looking at is the $\infty$-category of presheaves of spaces $\mathcal{P}(\mathcal{M}^{\mathrm{op}}) := \operatorname{Fun}(\mathcal{M}, \mathcal{S})$ on $\mathcal{M}^{\mathrm{op}}$, which is again presentable. There are a couple of different model categories that present $\mathcal{P}(\mathcal{M}^{\mathrm{op}})$, one of them being the projective model structure on a category of simplicial presheaves. You can find the details in section 5.1.1 of Lurie's *Higher Topos Theory*.


The $\infty$-category $\operatorname{Fun}^R(\mathcal{M}, \mathcal{S})$ of limit-preserving functors is equivalent to $\mathcal{M}^{\mathrm{op}}$ itself through the Yoneda embedding $\mathcal{M}^{\mathrm{op}} \to \mathcal{P}(\mathcal{M}^{\mathrm{op}})$; i.e., limit-preserving functors $\mathcal{M} \to \mathcal{S}$ are corepresentable. This is proposition 5.5.2.2 of *Higher Topos Theory*.


