Is there a natural way to give a bisimplicial structure on a 2-category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:30:29Z http://mathoverflow.net/feeds/question/5798 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5798/is-there-a-natural-way-to-give-a-bisimplicial-structure-on-a-2-category Is there a natural way to give a bisimplicial structure on a 2-category? Fei 2009-11-17T07:46:34Z 2009-11-17T12:15:38Z <p>I mean by the nerve functor. </p> <p>Given a 2-category $\mathcal{C}$, if we forget the 2-category structure (just view $\mathcal{C}$ as a category), the nerve functor will give us a simplicial set $N\mathcal{C}$. However, $\mathcal{C}$ is a 2-category, thus for any two objects $x,y\in\mathcal{C}$, $Hom_{\mathcal{C}}(x,y)$ is a category, applying the nerve functor gives us a simplicial set $N(Hom(x,y))$.</p> <p>My question is, can these two simplicial set structure compatible in some way, gives us a bisimplicial set $N_{p,q}(\mathcal{C})$, say? Or is there another way to give a bisimplicial structure on a 2-category? </p> http://mathoverflow.net/questions/5798/is-there-a-natural-way-to-give-a-bisimplicial-structure-on-a-2-category/5812#5812 Answer by Urs Schreiber for Is there a natural way to give a bisimplicial structure on a 2-category? Urs Schreiber 2009-11-17T12:15:38Z 2009-11-17T12:15:38Z <p>Yes. This is called the <a href="http://ncatlab.org/nlab/show/double+nerve" rel="nofollow">double nerve</a> of a 2-category.</p> <p>See in particular the first reference cited at that link.</p>