nLab
categorical wreath product

for disambiguation see wreath product

Definition

Definition

Let $A$ be a small category. Its categorical wreath product with the simplex category is the category $\Delta \wr A$ whose

• objects are $k$-tuples $([k],(a_1,\cdots ,a_k))$ of objects of $A$, for any $k\in \mathbb{N}$;

• morphisms are tuples

  $$(\varphi ,{\varphi }_{ij}):([k],(a_1,\cdots ,a_k))\to ([l],(b_1,\cdots ,b_l))$$(\phi, \phi_{i j}) : ([k],(a_1, \cdots, a_k)) \to ([l],(b_1, \cdots, b_l))

  consisting of

  • a morphism $\varphi :[k]\to [l]$ in $\Delta$;

  • morphisms ${\varphi }_{ij}:a_i\to b_j$ for $0<i\le k$ and $\varphi (i-1)<j\le \varphi (i)$.

(Berger, def. 3.1).

Remark

An object of $\Delta \wr A$ is to be thought of as a sequence of morphisms labeled by objects of $A$

$$\begin{array}{c}0\\ \downarrow a_1\\ 1\\ \downarrow a_2\\ \downarrow \\ ⋮\\ \downarrow a_n\\ n\end{array}$$\array{ 0 \\ \downarrow \mathrlap{a_1} \\ 1 \\ \downarrow \mathrlap{a_2} \\ \downarrow \\ \vdots \\ \downarrow \mathrlap{a_n} \\ n }

and morphisms are given by maps between these linear orders equipped with morphisms from the $k$th object in the source to all the objects in the target that sit in between the image of the $k$th step.

Examples

• The $k$-fold wreath product of the simplex category with itself is the $n$th Theta-category ${\Theta }_n={\Delta }^{\wr n}$.

• Other applications are discussed at club and at terminal coalgebra of an endofunctor.

References

Section 3 of

• Clemens Berger, Iterated wreath product of the simplex category and iterated loop spaces (arXiv:math/0512575),