Model for the (infinity,1)-category of (homotopy-)limit preserving functors

Question

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.

Answer 1

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.

Answer 2

Even if 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, it is not true that the $\infty$-category of presheaves of spaces $\mathcal{P}(\mathcal{M}^{\mathrm{op}}) := \operatorname{Fun}(\mathcal{M}, \mathcal{S})$ on $\mathcal{M}^{\mathrm{op}}$ is again presentable. This only holds if $\mathcal{M}$ is small which is typically not true for a model category. Also the results of Section 5.1.1 of Lurie's Higher Topos Theory giving a model for the $\infty$-category of presheaves only speak about the small case.

In the case you are interested in one should consider only small functors from $\mathcal{M}$ to $\mathcal{S}$ in order to get an $\infty$-category with small mapping spaces. This is what Lurie does when considering the pro category of spaces as a full subcategory of the $\infty$-category of accessible functors from spaces to spaces. (See Definition 7.1.6.1 in Higher Topos Theory. In this case small is equivalent to accessible.)

You should thus begin with the category of small simplicial functors from $M$ to simplicial sets, and endow it with the projective model structure that was constructed by Chorny and Dwyer: http://arxiv.org/abs/math/0607117. Then you should left Bousfield localize this model structure in such a way that the fibrant objects in the localized model structure would be the projectively fibrant functors that preserve homotopy limits. Note that you cannot use the usual theorems on Bousfield localizations such as those of Hirschhorn or Smith-Lurie because your model category is not cellular nor combinatorial (in fact, it is not even cofibrantly generated). You will have to use techniques such as in http://arxiv.org/abs/1409.8525, which rely on the generalized small object argument due to Chorny.

The localized model structure above would indeed model the $\infty$-category $\operatorname{Fun}^R(\mathcal{M}, \mathcal{S})$ of limit-preserving functors, which is equivalent to $\mathcal{M}^{\mathrm{op}}$ itself through the Yoneda embedding by proposition 5.5.2.2 of Higher Topos Theory, as Alberto García-Raboso mentioned.