What properties should a good definition of (weak) $n$-category satisfy? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:11:02Z http://mathoverflow.net/feeds/question/49916 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49916/what-properties-should-a-good-definition-of-weak-n-category-satisfy What properties should a good definition of (weak) $n$-category satisfy? Daniel Litt 2010-12-19T23:29:55Z 2010-12-20T11:36:37Z <p>My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted. I've run across several properties such definitions should satisfy, e.g. that weak $n$-groupoids should model homotopy $n$-types, or that the collection of $n$-categories should naturally be an $n+1$-category. But I haven't found anything like a comprehensive list of conjectural properties a "good" theory of weak $n$-categories should satisfy.</p> <blockquote> <p>Is there such a list somewhere in the literature?</p> </blockquote> <p>If not, I'd of course be thrilled with an answer containing such a list.</p> http://mathoverflow.net/questions/49916/what-properties-should-a-good-definition-of-weak-n-category-satisfy/49917#49917 Answer by Andy Putman for What properties should a good definition of (weak) $n$-category satisfy? Andy Putman 2010-12-19T23:44:00Z 2010-12-20T00:26:15Z <p>I'm far from an expert on this subject, but the book "Higher-Dimensional Categories: an illustrated guide book" by Cheng and Lauda (available <a href="http://cheng.staff.shef.ac.uk/guidebook/index.html" rel="nofollow">here</a>) has a very nice description of the things that various competing notions of n-categories are trying to generalize and the motivations behind them. It's not quite a cut-and-dried "list of properties", but my impression is that the philosophies behind the various definitions are too complicated to just make a single list.</p> http://mathoverflow.net/questions/49916/what-properties-should-a-good-definition-of-weak-n-category-satisfy/49936#49936 Answer by André Henriques for What properties should a good definition of (weak) $n$-category satisfy? André Henriques 2010-12-20T07:58:05Z 2010-12-20T07:58:05Z <p>I've just heard a talk by Julia Bergner on the subject, reporting on her ongoing work with Charles Rezk. Here's the list of properties that they are trying to establish for their model of $(\infty,n)$-categories. </p> <ul> <li><p>For each $n>0$, the category of $(\infty,n)$-categories is a cartesian model category (i.e. it's a model category, and it's compatible with cartesian product).</p></li> <li><p>The category of $(\infty,0)$-categories is Quillen equivalent to the category of topological spaces.</p></li> <li><p>For any $n\ge 0$, the category of $(\infty,n+1)$-categories is Quillen equivalent to the category of categories enriched over $(\infty,n)$-categories.</p></li> </ul> <p>Clearly, not every model for $(\infty,n)$-categories needs to comply to the above list of properties. But it's also quite clear that <em>if</em> a model for $(\infty,n)$-categories complies to the above requirements, then it's a good model.</p> <p><hr> And just for the record, an $(\infty,n)$-category is supposed to capture the idea of an $\infty$-category, all of whose $k$-moorphisms are invertible for $k>n$.</p> http://mathoverflow.net/questions/49916/what-properties-should-a-good-definition-of-weak-n-category-satisfy/49938#49938 Answer by David Roberts for What properties should a good definition of (weak) $n$-category satisfy? David Roberts 2010-12-20T08:19:34Z 2010-12-20T08:19:34Z <p>To complement Andy's answer, there is also Tom Leinster's <a href="http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html" rel="nofollow">A survey of definitions of n-category</a>, which has a good 'further reading' section. He also emphasises that there are many other interesting objects in higher category theory, and my guess is that these provide examples for, and non-trivial extensions of, $n$-categories.</p> http://mathoverflow.net/questions/49916/what-properties-should-a-good-definition-of-weak-n-category-satisfy/49954#49954 Answer by Todd Trimble for What properties should a good definition of (weak) $n$-category satisfy? Todd Trimble 2010-12-20T11:36:37Z 2010-12-20T11:36:37Z <p>Also read some of the original papers by Baez and Dolan, for example Categorification (in Higher Category Theory, Contemporary Mathematics 230, AMS 1998), where the Tangle Hypothesis is explained among other things. </p> <p>It is hoped that a satisfactory definition on n-category would provide a framework for proving the homotopy hypothesis, the tangle hypothesis, and the stabilization conjecture, as discussed at the n-Category Caf&eacute; <a href="http://golem.ph.utexas.edu/category/2007/05/questions_on_ncategories_and_t.html" rel="nofollow">here</a>. </p> <p>I wouldn't say that none of the proposed definitions is generally accepted, but rather that none has emerged as clearly the best definition. In some sense all are accepted as capturing some useful intuition, although some seem easier to work with than others for some specific purpose. Perhaps the most developed notions are the more geometric ones which take advantage of decades of work in homotopy theory. </p>