nLab wreath product of wreaths

For disambiguation see at wreath product.

Idea

Wreaths are the natural generalization of distributive laws. Like the latter, they produce a sort of composite monad, the wreath product.

Formally, let $EM(\mathcal{K})$ be the free completion of a 2-category under $EM$ objects. A wreath is an object in $EM(EM(\mathcal{K}))$, and since $EM$ is a monad, it admits a multplication which is indeed the wreath product $\wr : EM(EM(\mathcal{K})) \to EM(\mathcal{K})$.

Definition

Let $(A,(t,\eta,\mu),(s, \lambda, \sigma, \nu))$ be a wreath in $\mathcal{K}$. Its wreath product is the monad on $A$ in $\mathcal{K}$ so defined:

1. its carrier 1-cell is $st$ (first $t$ then $s$),
2. its unit is $\sigma : 1 \to st$,
3. its multiplication is
   $$stst \xrightarrow{s \lambda t} sstt \xrightarrow{ss \mu} sst \xrightarrow{\nu t} stt \xrightarrow{s\mu} st$$

Remark

Every distributive law is a wreath, and the wreath product of a distributive law qua wreath indeed coincides with the composite monad one would get out of the distributive law.

Related concepts

• wreath
• weak wreath product

References

• Stephen Lack, Ross Street, The formal theory of monads II, Journal of Pure and Applied Algebra 175 1-3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]