ScalingStacks

Definition 6.1 ([9, Definition 3.1]). Let $C$ be a small category. The wreath product $\Delta\wr C$ is the category

• •

  whose objects consist of tuples $([n];c_{1},\dots,c_{n})$ where $[n]\in\Delta$ and $c_{i}\in C$, and

• •

  whose morphisms from $([m];a_{1},\dots,a_{m})$ to $([n];b_{1},\dots,b_{n})$ consist of tuples $(\phi;\phi_{ij})$, where $\phi:[m]\to[n]$, and $\phi_{ij}:a_{i}\to b_{j}$ where $0<i\leq m$, and $\phi(i-1)<j\leq\phi(i)$.

The category $\Theta_{n}$ is now defined inductively as a wreath product: $\Theta_{1}=\Delta$, and $\Theta_{n}=\Delta\wr\Theta_{n-1}$. In particular this gives rise to embeddings $\sigma:\Theta_{n-1}\to\Theta_{n}$, given by $\sigma(o)=([1];o)$, and $\iota:\Delta\to\Theta_{n}$ given by $\iota([n])=([n];([0]),\dots,([0]))$.