Berger-Joyal duality and traces I

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