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This question is a follow up to: Model for the (infinity,1)-category of (homotopy-)limit preserving functors.

Warm-up Question: Given a simplicial model category $M$, what model category models the $(\infty, 1)$-category of presheaves of spaces on the $(\infty,1)$-category associated to $M$?

I'm skeptical that the projective/injective model structures on simplicial presheaves on $M$ achieve this goal because they don't seem to use the weak equivalences of $M$ at all. (Although now that I think about maybe SSet-enriched functors "see" the weak equivalences in $M$.)

I'll use $N^{hc}(M^{cf})$ to denote the homotopy-coherent nerve of the simplicial subcategory spanned by the fibrant-cofibrant objects, ie the $(\infty,1)$-category associated to $M$.

Question: Given a simplicial model category $M$ and a fixed diagram category $D$, what model category models the $(\infty,1)$-category of functors from $N^{hc}(M^{cf})$ to Spaces which preserve homotopy limits indexed by $D$?

I was hoping the answer would look something like the following. Denote by Fun(M,SSet) the model category which answers the warm-up question, and by S the collection of natural transformations {F(hlim X) ---> hlim FX } where S ranges over $F:M \to \textrm{Spaces}, X: D \to M$. Then the model category of D-shaped homotopy limit preserving functors from $M$ to Spaces is modeled by the (right?) Bousfield localization of Fun(M,SSet) by S.

If you do answer the question the way I hoped, please say something mildly conciliatory about the fact that S seems too big.

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Oh this isn't right: the maps F(hlimX) ---> hlim FX aren't maps of functors. That's embarrassing. – Joey Hirsh Feb 19 '13 at 23:02
up vote 4 down vote accepted

The $(\infty,1)$-category of presheaves on any small $(\infty,1)$-category $C$ is presented by the model structure of simplicial presheaves on any simplicial category which incarnates $C$. So if $M$ is small, then simplicial presheaves on the simplicial category $M^{cf}$ would do it, or on the hammock localization of $M$, or any other weakly equivalent simplicial category.

If $M$ is not small, then you can pass to a higher universe in which it is. I'm a little doubtful that there is a good model category which presents presheaves on a large domain that take small values, although you might be interested in this paper.

For the second question, I think you want the left Bousfield localization, but other than that your idea is correct (once you deal with size issues as above, so that $S$ is small).

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Can you say a little more about passing to larger universes with regard to model categories? This is not a precise question, but when it comes to model categories I worry that there are problems with passing to larger universes that I can't see, like maybe something to do with the small object argument. I guess what I'm asking for is a math-statement like "For a given universe U, and a simplicial model U-category M, there is a universe U' s.t U \subset U' and Fun_{U-SSet}(M, U-SSet) is a simplicial model U'-category." Is that true? – Joey Hirsh Jan 8 '13 at 16:10
...and by model category let's say I want functorial factorizations. – Joey Hirsh Jan 8 '13 at 16:22
Your statement is true (or, at least, follows from the axiom of universes) if you meant Fun_{U-SSet}(M, U'-SSet), if we take U' sufficiently large that M is U'-small. There's no way you're going to get a U'-complete category by taking functors into a category (like U-SSet) that's only U-complete. – Mike Shulman Jan 9 '13 at 9:25
Great. Thanks Mike. – Joey Hirsh Jan 9 '13 at 15:42

You should begin with the projective model structure on the category of small simplicial functors from $M^{op}$ to simplicial sets due to Chorny and Dwyer This models the $\infty$-category of small functors from the underlying $\infty$-category of $M^{op}$ to spaces. Since $M$ is usually large, you must restrict to small functors in order to get small mapping spaces in the $\infty$-category of presheaves on $M$. Then you should continue as in my answer to Model for the (infinity,1)-category of (homotopy-)limit preserving functors.

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