Presheaves on a complete Segal space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:12:31Z http://mathoverflow.net/feeds/question/82685 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82685/presheaves-on-a-complete-segal-space Presheaves on a complete Segal space Mike Shulman 2011-12-05T08:36:44Z 2011-12-06T16:39:14Z <p>Let <em>C</em> be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over <em>C</em> which presents the $(\infty,1)$-presheaf category of <em>C</em>? Ideally, such a model structure would be Quillen equivalent to the contravariant model structure over a quasicategory incarnation of <em>C</em>, and to the projective model structure for simplicial presheaves on a simplicial-category incarnation of <em>C</em>.</p> http://mathoverflow.net/questions/82685/presheaves-on-a-complete-segal-space/82711#82711 Answer by Charles Rezk for Presheaves on a complete Segal space Charles Rezk 2011-12-05T15:57:25Z 2011-12-06T16:39:14Z <p>Yes. Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W_q)$. The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operator $\delta:[q]\to [p]$ with $\delta(q)=p$, the evident map from $X_p$ to the pullback of $$X_q \xrightarrow{f} W_q \xleftarrow{\delta} W_p$$ is a weak equivalence of spaces. (<strong>Edit:</strong> in fact, it suffices to require the evident map to the pullback to be a weak equivalence only for $\delta:[0]\to[p]$ with $\delta(0)=p$.)</p> <p>I worked out some of this years ago, but never finished it; somebody should do this (or perhaps someone has already?). Lurie has done pretty much exactly the same thing in the context of quasi-categories, in HTT.</p>