for disambiguation see wreath product
[[!include category theory - contents]]
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
consisting of
a morphism $\phi: [k] \to [l]$ in $\Delta$;
morphisms $\phi_{i j} : a_i \to b_j$ for $0 \lt i \leq k$ and $\phi(i-1) \lt j \leq \phi(i)$.
An object of $\Delta \wr A$ is to be thought of as a sequence of morphisms labeled by objects of $A$
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.
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.
Section 3 of
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