Model for the (infinity,1)-category of functors preserving certain homotopy limits - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:15:04Z http://mathoverflow.net/feeds/question/117304 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117304/model-for-the-infinity-1-category-of-functors-preserving-certain-homotopy-limit Model for the (infinity,1)-category of functors preserving certain homotopy limits Joey Hirsh 2012-12-27T05:11:40Z 2013-01-07T07:57:32Z <p>This question is a follow up to: <a href="http://mathoverflow.net/questions/117267/model-for-the-infinity-1-category-of-homotopy-limit-preserving-functors" rel="nofollow">http://mathoverflow.net/questions/117267/model-for-the-infinity-1-category-of-homotopy-limit-preserving-functors</a>.</p> <blockquote> <p>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 $M$? </p> </blockquote> <p>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 $M$.)</p> <p>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 $M$.</p> <blockquote> <p>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 $D$?</p> </blockquote> <p>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.</p> <p>If you do answer the question the way I hoped, please say something mildly conciliatory about the fact that S seems too big.</p> http://mathoverflow.net/questions/117304/model-for-the-infinity-1-category-of-functors-preserving-certain-homotopy-limit/118250#118250 Answer by Mike Shulman for Model for the (infinity,1)-category of functors preserving certain homotopy limits Mike Shulman 2013-01-07T07:57:32Z 2013-01-07T07:57:32Z <p>The $(\infty,1)$-category of presheaves on any <em>small</em> $(\infty,1)$-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.</p> <p>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 <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.hha/1355321074" rel="nofollow">this paper</a>.</p> <p>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).</p>