Berger-Joyal duality and traces I

Abstract

We give a new proof that the opposite of Joyal's disk category ${\mathcal{D}}_n$ is Berger's wreath product category $\hat{I}{˜}_n=\hat{I}”\wr \cdots \wr \hat{I}”$. Our techniques continue to apply when the simplex category $\hat{I}”$ is replaced by Connes' cyclic category $\hat{I}›$ and some other crossed simplicial groups.