What's the name of this flavor of n-category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:25:12Z http://mathoverflow.net/feeds/question/45782 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45782/whats-the-name-of-this-flavor-of-n-category What's the name of this flavor of n-category? Kevin Walker 2010-11-12T03:34:16Z 2010-11-13T16:41:58Z <p>I'm looking for the name of a certain n-category definition. (Someone explained it to me a couple of years ago. I remember the definition, but not the name. Without the name it's difficult to search for a citation. I want the citation in order to explain something we're <em>not</em> doing in a paper.)</p> <p>For background, consider the Moore loop space $\Omega_r$ of loops of length $r$ (that is, parameterized by the interval $[0,r]$). We have a strictly associative composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. The main idea of an "xxxx" n-category is to imitate this idea in higher dimensions. The $k$-morphisms are parameterised by $k$-dimensional rectangles with sides of lengths $r_1,\ldots,r_k$. Gluing rectangles together gives $k$ different strictly associative ways to compose $k$-morphisms.</p> <p>Question: What is "xxxx" above?</p> <p>Bonus question: What's the best (or any) citation for this idea?</p> <hr> <p>EDIT: It turns out the definition I was trying to remember is unpublished work of Ulrike Tillmann. But the version from Ronnie Brown linked to in David Roberts' answer is pretty similar (for my purposes, at least).</p> http://mathoverflow.net/questions/45782/whats-the-name-of-this-flavor-of-n-category/45783#45783 Answer by David Roberts for What's the name of this flavor of n-category? David Roberts 2010-11-12T03:54:12Z 2010-11-12T03:54:12Z <p>Ronnie Brown has a related idea, contained in this article:</p> <blockquote> <p>Moore hyperrectangles on a space form a strict cubical omega-category<br/> <a href="http://arxiv.org/abs/0909.2212" rel="nofollow">arXiv</a></p> </blockquote> <p>discussed briefly <a href="http://ncatlab.org/nlab/show/Moore+path+category" rel="nofollow">here at the nLab</a>.</p> <p>If you are instead thinking of a globular $n$-category, the closest I know of is a <a href="http://ncatlab.org/nlab/show/Trimble+n-category" rel="nofollow">Trimble n-category</a>, but that doesn't use Moore paths, but paths of length 1 and the $A_\infty$-co-category structure on $[0,1]$.</p> http://mathoverflow.net/questions/45782/whats-the-name-of-this-flavor-of-n-category/45800#45800 Answer by Peter LeFanu Lumsdaine for What's the name of this flavor of n-category? Peter LeFanu Lumsdaine 2010-11-12T08:10:17Z 2010-11-12T15:08:15Z <p><strong>Simpson-semistrict</strong> $n$-categories could be what you're after: $n$-categories where everything except the unit laws holds strictly, generalising one of the crucial properties of Moore path spaces? It's not a specific definition of $n$-category, but a strictness property which can be applied within various definitions.</p> <p>Carlos Simpson has conjectured that these are enough to model homotopy types; Moore path space show this in dimension 1. I know very little about the details of this myself, I'm afraid, but what I have read about it is mostly from these sources plus their links and discussions:</p> <ul> <li>Simpson, <a href="http://arxiv.org/abs/math/9810059" rel="nofollow">Homotopy types of strict 3-groupoids</a>.</li> <li>nlab: <a href="http://ncatlab.org/nlab/show/semi-strict+infinity-category" rel="nofollow">semi-strict $\infty$-category</a></li> <li>nlab: Simpson’s conjecture (I can't figure out how to link this directly; the single-quote in the url seems to confuse markdown)</li> <li>n-Category Café: Urs Schreiber, <a href="http://golem.ph.utexas.edu/category/2008/10/semicategories.html" rel="nofollow">Semistrict Infinity-Categories and ω-Semi-Categories</a></li> </ul> <p>I believe several people have been making some progress on it recently; eg Makkai mentioned some results along these lines at the latest <a href="http://www.mscs.dal.ca/~selinger/ofest2010/" rel="nofollow">Octoberfest</a>.</p>