Is the $\infty$-category of stable $\infty$-categories stable? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:14:24Z http://mathoverflow.net/feeds/question/59270 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59270/is-the-infty-category-of-stable-infty-categories-stable Is the $\infty$-category of stable $\infty$-categories stable? Alex 2011-03-23T08:23:59Z 2011-03-23T18:46:44Z <p>More generally, are there any remarkable properties enjoyed by the $\infty$-category of stable $\infty$-categories?</p> http://mathoverflow.net/questions/59270/is-the-infty-category-of-stable-infty-categories-stable/59342#59342 Answer by Dustin Clausen for Is the $\infty$-category of stable $\infty$-categories stable? Dustin Clausen 2011-03-23T18:46:44Z 2011-03-23T18:46:44Z <p>No. It's pointed by the zero category, but then taking loops of a stable category C (in the sense of the pullback of 0 --> C along itself) always gives the zero category, so loops is definitely not an equivalence.</p> <p>One important structural feature of the category of stable categories along these lines is that it has some nice cofiber sequences (Verdier localization sequences), but I'm not sure the categorical properties these satisfy have been axiomatized (into a possible definition of stable (infty,2)-category''?)</p>