Nerves of (braided or symmetric) monoidal categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:11:46Z http://mathoverflow.net/feeds/question/35097 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35097/nerves-of-braided-or-symmetric-monoidal-categories Nerves of (braided or symmetric) monoidal categories domenico fiorenza 2010-08-10T10:56:14Z 2010-08-11T07:19:41Z <p>I'm looking for references on the structure which can be roughtly described as follows: given a (braided or symmetric) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with a single vertex, an edge for every object of $C$, a triangle with edges $X,Y,Z$ for every morphism $\varphi:Z\to X\otimes Y$, a tethraedron for every four triangles making up a commutative diagram involving the associator of $C$, higher coherences..</p> <p>Any suggestion? thanks</p> http://mathoverflow.net/questions/35097/nerves-of-braided-or-symmetric-monoidal-categories/35128#35128 Answer by S. Carnahan for Nerves of (braided or symmetric) monoidal categories S. Carnahan 2010-08-10T15:31:16Z 2010-08-10T15:31:16Z <p>If you want to capture the structure of the category together with its monoidal structure, you may need a $k$-fold simplicial set for $k>1$, i.e., a functor from $(\Delta^{op})^k$ to sets. One of the simplicial coordinates encodes the composition law in the category, another encodes the monoidal structure, and the rest decribe compatibility between monoidal structures (if the monoidal structure is braided or symmetric). See also <a href="http://ncatlab.org/nlab/show/double+nerve" rel="nofollow">Double nerve</a>. You may want to look up work by Baez and Dolan on their <a href="http://ncatlab.org/nlab/show/periodic+table" rel="nofollow">periodic table</a> that expresses monoidal categories of various types as higher categories with connectedness properties. In particular, there is an equivalence between monoidal categories and 2-categories with one object, and an equivalence between braided monoidal categories and 3-categories with one object and one 1-morphism.</p> http://mathoverflow.net/questions/35097/nerves-of-braided-or-symmetric-monoidal-categories/35172#35172 Answer by Omar Antolín-Camarena for Nerves of (braided or symmetric) monoidal categories Omar Antolín-Camarena 2010-08-10T21:32:58Z 2010-08-10T21:43:13Z <p>For plain old monoidal categories, you could regard them as a bicategory with a single object and use the <a href="http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html" rel="nofollow">Duskin nerve</a>. For braided or symmetric categories there might be higher nerves you can take, but I'm not sure how well those work. You might be better off using the k-fold simplicial sets that Scott suggested.</p>