# nLab categorical wreath product

for disambiguation see wreath product

category theory

# Contents

## 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 $\left(\left[k\right],\left({a}_{1},\cdots ,{a}_{k}\right)\right)$ of objects of $A$, for any $k\in ℕ$;

• morphisms are tuples

$\left(\varphi ,{\varphi }_{ij}\right):\left(\left[k\right],\left({a}_{1},\cdots ,{a}_{k}\right)\right)\to \left(\left[l\right],\left({b}_{1},\cdots ,{b}_{l}\right)\right)$(\phi, \phi_{i j}) : ([k],(a_1, \cdots, a_k)) \to ([l],(b_1, \cdots, b_l))

consisting of

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

• morphisms ${\varphi }_{ij}:{a}_{i}\to {b}_{j}$ for $0 and $\varphi \left(i-1\right).

###### Remark

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

$\begin{array}{c}0\\ ↓{a}_{1}\\ 1\\ ↓{a}_{2}\\ ↓\\ ⋮\\ ↓{a}_{n}\\ n\end{array}$\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.

## References

Section 3 of

Revised on February 12, 2013 17:02:57 by Manuel Baerenz (128.243.253.112)