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

2012-12-27T05:11:40Z

This question is a follow up to: http://mathoverflow.net/questions/117267/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.

Answer by Mike Shulman for Model for the (infinity,1)-category of functors preserving certain homotopy limits

2013-01-07T07:57:32Z

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).

Model for the (infinity,1)-category of functors preserving certain homotopy limits - MathOverflow

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

Answer by Mike Shulman for Model for the (infinity,1)-category of functors preserving certain homotopy limits