Candidate definitions for "1-braided 2-category"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:54:59Z http://mathoverflow.net/feeds/question/13590 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13590/candidate-definitions-for-1-braided-2-category Candidate definitions for "1-braided 2-category"? Theo Johnson-Freyd 2010-01-31T22:17:42Z 2010-02-01T06:03:10Z <p>Recall that a <em>braided monoidal category</em> is a category $\mathcal C$, a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$ satisfying some properties, and a natural isomorphism $b_{V,W}: V\otimes W \to W\otimes V$ satisfying some properties. Recall also that a <em>monoidal category</em> (just $\mathcal C,\otimes$ and their properties) is the same as a one-element 2-category: the objects of $\mathcal C$ become the morphisms, and the monoidal structure becomes composition.</p> <p>Thus, is there a natural definition of "1-braided 2-category"? I'm calling it "1-braided" because the braiding acts on 1-morphisms (as opposed to "braided monoidal 2-category", where the braiding acts on the 0-morphisms).</p> <p>I realize, of course, that if $V,W$ are morphisms of a 2-category so that $V\circ W$ is defined, then generally $W\circ V$ is not defined, so a priori asking for any relationship $V\circ W \cong W\circ V$ doesn't make sense. On the other hand, consider <a href="http://arxiv.org/abs/0803.3652" rel="nofollow">Aaron Lauda's categorification of $U_q(\mathfrak{sl}_2)$</a>. It is a 2-category, but different hom-sets can be more-or-less identified.</p> http://mathoverflow.net/questions/13590/candidate-definitions-for-1-braided-2-category/13602#13602 Answer by Mike Shulman for Candidate definitions for "1-braided 2-category"? Mike Shulman 2010-01-31T23:41:53Z 2010-01-31T23:41:53Z <p>You might be interested in <a href="http://arxiv.org/abs/0910.1306" rel="nofollow">this paper</a>. While the notion of "bicategory with a shadow" is not exactly what you're describing, I think it might be at least related. It is, at least, a structure on a bicategory which generalizes the notion of symmetry on a monoidal category. (It <em>doesn't</em> generalize braidings that aren't symmetries, though.)</p> http://mathoverflow.net/questions/13590/candidate-definitions-for-1-braided-2-category/13624#13624 Answer by S. Carnahan for Candidate definitions for "1-braided 2-category"? S. Carnahan 2010-02-01T04:40:28Z 2010-02-01T04:40:28Z <p>As you mentioned, the commutor doesn't generally make sense if your 2-category has morphisms between distinct objects. Therefore, here's a first try at a definition: a collection of braided monoidal categories.</p> <p>There is another perspective: you can fold the commutor into the structure of a 3-category with one object and one 1-morphism instead of describing it as an extra datum. Here's a more refined candidate definition: a 3-category with one object.</p> http://mathoverflow.net/questions/13590/candidate-definitions-for-1-braided-2-category/13629#13629 Answer by Noah Snyder for Candidate definitions for "1-braided 2-category"? Noah Snyder 2010-02-01T06:03:10Z 2010-02-01T06:03:10Z <p>In the Baez-Dolan periodic table a braided monoidal category is just a 2-monoidal category (that is a 3-category with one object and one 1-morphism). If you just think that a braiding means that the structure is inherently 3-dimensional, then you might just want to think about a 1-monoidal 2-category (that is a 3-category with one object).</p>