nLab categorical wreath product

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for disambiguation see wreath product

category theory

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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

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

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 $\phi(i-1) \lt j \leq \phi(i)$.

Remark

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

$\array{ 0 \\ \downarrow \mathrlap{a_1} \\ 1 \\ \downarrow \mathrlap{a_2} \\ \downarrow \\ \vdots \\ \downarrow \mathrlap{a_n} \\ n }$

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

Last revised on February 12, 2013 at 17:02:57. See the history of this page for a list of all contributions to it.