truncation commutes with localization? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T15:48:17Z http://mathoverflow.net/feeds/question/69677 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69677/truncation-commutes-with-localization truncation commutes with localization? euklid345 2011-07-06T22:17:16Z 2011-09-01T16:00:01Z <p>Suppose I have a higher category, say a simplicial category $C$, and I want to invert a certain type of morphism $W$. Simplicial localization (for example hammock localization), gives a localized simplicial categetory $W^{-1}C$. Then I truncate this category by passing to $\pi_0$ of hom-sets to obtain the 1-category $\pi_0(W^{-1}C)$. </p> <p>Conversely, I may pass to the truncated 1-category $\pi_0(C)$, and then localize, to get the 1-category $W^{-1}\pi_0(C)$. </p> <p>I guess there is a functor $\pi_0(W^{-1}C)\to W^{-1}\pi_0(C)$. Is it an equivalence of categories? I am also interested in 2-category truncations defined by $\Pi_1$ of hom-sets. </p> <p>If it is not true, are there easy to understand counterexamples? </p> http://mathoverflow.net/questions/69677/truncation-commutes-with-localization/74268#74268 Answer by euklid345 for truncation commutes with localization? euklid345 2011-09-01T16:00:01Z 2011-09-01T16:00:01Z <p>This is Denis-Charles Cisinski's answer, given in the comments:</p> <p>Yes, the map $\pi_0(W^{-1}C)\to W^{-1}\pi_0(C)$ is always an equivalence of categories (this follows immediately by comparing the corresponding universal properties). The same remains true (for formal reasons as well) if you look at the truncations of hom-spaces defined by $\pi_1$ and work up to an adequate notion of equivalence of 2-categories (in fact, you may as well truncate in dimension $n$ and get an equivalence of (n+1,1)-categories).</p>