local model structure on simplicial presheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:07:37Z http://mathoverflow.net/feeds/question/91618 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91618/local-model-structure-on-simplicial-presheaves local model structure on simplicial presheaves Sasha 2012-03-19T13:30:20Z 2012-03-19T16:29:32Z <p>Hello,</p> <p>Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.</p> <p>Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its projective model structure (fib. and w.e. are level-wise).</p> <p>Then one defines the class $S$ of local w.e. to be that of some maps of simplicial presheaves which induce isomorphisms on homotopy groups etc...</p> <p>Then one takes the left Bousfield localization of the projective model structure along $S$, to get the projective local model structure (that which models "homotopy" sheaves).</p> <p>I don't understand much in this things, so I have two questions:</p> <blockquote> <p>1) In general, given a set $S$ of maps, we define the set of $S$-local equivalences (those which satisfy some left property w.r.t. $S$-local objects, which are those which satisfy some right property w.r.t. $S$...). For our $S$, will $S$-local equivalences coincide with $S$?</p> <p>2) If I take $T$ to be the set of hypercovers, will $T$-local equivalences be $S$?</p> </blockquote> <p>Thank you very much, Sasha</p> http://mathoverflow.net/questions/91618/local-model-structure-on-simplicial-presheaves/91625#91625 Answer by Alberto García-Raboso for local model structure on simplicial presheaves Alberto García-Raboso 2012-03-19T15:11:52Z 2012-03-19T16:29:32Z <p>The answers to both of your questions is yes.</p> <p>The original description by Jardine (<a href="http://www.ams.org/mathscinet-getitem?mr=906403" rel="nofollow">Simplicial presheaves</a>, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87) of the local injective model structure on simplicial presheaves <em>defines</em> the weak equivalences as the class you refer in your question as $S$. This means that Jardine checked explicitly that the category of simplicial presheaves with the class $S$ of weak equivalences and the class of global cofibrations does indeed satisfy all the axioms in the definition of a model category.</p> <p>Notice that given two model category structures $(W, cof, fib)$ and $(W', cof', fib')$ on the same underlying category $C$, you say that the second is a <a href="http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories" rel="nofollow">left Bousfield localization</a> of the first if $cof = cof'$ and $W \subseteq W'$, and that the above picture fits this definition.</p> <p>Now, given a model category structure $(W, cof, fib)$ and an arbitrary class $M$ of morphisms, you can try to define a new model category structure $(W', cof', fib')$ which is a left Bousfield localization of the first and which is minimal subject to the condition that $M \subseteq W'$. Then $W'$, the class of so-called $M$-local equivalences, is a kind of saturation of $M$, in the sense that it is the smallest class of morphisms containing $M$ and satisfying the necessary conditions for it to be the class of weak equivalences of a model category structure on your category (with the cofibrations fixed, of course). Since your class $S$ is already saturated in this sense, it is itself the class of $S$-local equivalences.</p> <p>Your second question is precisely the beautiful paper <a href="http://www.math.uiuc.edu/K-theory/0563/" rel="nofollow">Hypercovers and simplicial presheaves</a> (Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51) by Dugger, Hollander and Isaksen.</p> <p>A nice guide to everything that is going on is at the nLab page <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves" rel="nofollow">model structure on simplicial presheaves</a>.</p>