nLab wreath product of wreaths

Idea

A wreath is a natural generalization of distributive law. Like the latter, it produces 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

• Steve Lack, Ross Street, The formal theory of monads II, Special volume celebrating the 70th birthday of Professor Max Kelly.

  J. Pure Appl. Algebra 175 (2002), no. 1-3, 243–265.