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

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

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

(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 $A$

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

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

• Michael Barr, Charles Wells, Section 12.4 of: Category theory for computing science, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories 22 (2012) 1-538 [pdf, tac:tr22]

See also:

• Charles Wells: A Krohn-Rhodes theorem for categories, J. Algebra 64 1 (1980) 37-45 [doi:10.1016/0021-8693(80)90130-1]

• Valdis Laan: Wreath product of set-valued functors and tensor multiplication, Semigroup Forum. 70 Springer (2005) [doi:10.1007/s00233-004-0162-9]

• Clemens Berger, Section 3 of: Iterated wreath product of the simplex category and iterated loop spaces, Advances in Mathematics 213 1 (2007) 230-270 [arXiv:math/0512575, doi:10.1016/j.aim.2006.12.006]