most recent 30 from http://mathoverflow.net 2013-05-25T12:50:25Z http://mathoverflow.net/feeds/question/3797 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3797/omega-categories-and-n-fold-complete-segal-spaces Michael Hoffman 2009-11-02T13:56:22Z 2009-11-02T16:57:31Z

<p>Why are n-fold complete segal spaces or (\infty, n)-categories (which I'm unsure of how to distinguish from omega-categories) important for n >= 3? Why are they "badly behaved" for n >= 3? (Lurie refers to them this way in his thesis).</p> <p>Also, I'm particularly interested to connections between n-fold complete segal spaces with regards to a question asked recently about "same" proofs. Is a 2-fold complete segal space sufficient in this particular arena?</p> <p>(Please tell me if this question is ill-posed, I'm just currently learning category theory)</p>

http://mathoverflow.net/questions/3797/omega-categories-and-n-fold-complete-segal-spaces/3805#3805 Charles Rezk 2009-11-02T15:35:54Z 2009-11-02T16:07:17Z

<p>n-fold complete Segal spaces are one <em>model</em> for (&infin;,n)-categories; there are other models. More precisely, they are supposed to be a model for <em>weak</em> (&infin;,n)-categories.</p> <p>The distinction that I think you are asking about is between <em>weak</em> and <em>strict</em>. Strict n-categories can be easily defined by a recursive definition: a strict n-category is just a category enriched over strict (n-1)-categories. A strict 1-category is just a plain-old category. Though easy to define, strict n-categories don't seem to capture the things people want an n-category to capture. </p> <p>One such feature is that strict n-categories don't satisfy the "homotopy hypothesis", which says that an n-groupoid (=n-category in which all morphisms are in some sense invertible) should model homotopy n-types (=spaces whose homotopy groups vanish above dimension n). In fact, this failure only occurs for n>=3; I believe this is the type of bad behavior Lurie refers to. Another failure of strict n-categories happens when you try to talk about higher monoidal structures.</p> <p>If you haven't already, take a look at the papers by Baez-Dolan on arxiv, which discuss a lot of these issues.</p>