most recent 30 from http://mathoverflow.net 2013-05-25T20:11:21Z http://mathoverflow.net/feeds/question/101499 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101499/are-pivotal-categories-the-algebras-for-a-cartesian-monad Bruce Westbury 2012-07-06T14:09:14Z 2012-07-11T21:27:05Z

<p>It seems to be "known" but not written down that the following are more-or-less equivalent:</p> <pre><code>pivotal (and spherical) categories spiders, as defined by Greg Kuperberg planar algebras, as defined by Vaughan Jones non-symmetric modular operads, following Getzler &amp; Kapranov </code></pre> <p>Then there is also the notion of a $T$-operad where $T$ is a cartesian monad. This was originally defined by Burroni and is the central concept in "Higher operads, higher categories" by MO contributor Tom Leinster.</p> <p>My question is wether the first example can also be understood using the general theory of $T$-operads?</p> <p>I don't have any good reason for asking. It is more that my head starts to hurt after contemplating this for any length of time.</p> <p>For a more specific question: there is a forgetful functor from pivotal categories to directed graphs. I believe this is monadic and that the associated monad is cartesian. If so, we can consider the associated theory of $T$-operads and $T$-multicategories. However, so far, I have not been able to penetrate this definition.</p>