nLab wreath

Idea

A wreath is a generalisation of a distributive law between two monads in a 2-category. While a distributive law in a 2-category $K$ can be seen as an object of $Mnd(Mnd(K))$, a wreath can be seen as an object of $EM(EM(K))$, where $EM$ denotes the completion of a 2-category under Eilenberg–Moore objects. Since $EM$ is a 2-monad, the multiplication $EM \circ EM \to EM$ produces from every wreath a composite monad, called its wreath product.

Definition

Let $\mathcal{K}$ be a 2-category. A wreath in $\mathcal{K}$ consists of:

1. A monad $(A,t, \eta, \mu)$ in $\mathcal{K}$,
2. A 1-cell $(s,\lambda):(A,t) \to (A,t)$ in $EM(\mathcal{K})$, that is a 1-cell $s:A \to A$ in $\mathcal{K}$ and a 2-cell $\lambda : ts \to st$ satisfying the axioms:

1. Two 2-cells $\sigma : 1 \to (s,\lambda)$ and $\mu : (s, \lambda)(s,\lambda) \to (s, \lambda)$ in $EM(\mathcal{K})$, which amount to two 2-cells in $\mathcal{K}$, $\sigma : 1 \to st$ and $nu : ss \to st$, such that

Duality

A cowreath (for lack of better terminology) in a 2-category $\mathcal{K}$ is an object of $KL(KL(\mathcal{K}))$, where $KL(\mathcal{K}) = EM(\mathcal{K}^{op})^{op}$ is the free Kleisli completion of $\mathcal{K}$.

It consists of

1. A monad $(A,t, \eta, \mu)$ in $\mathcal{K}$,
2. A 1-cell $(s,\lambda):(A,t) \to (A,t)$ in $KL(\mathcal{K})$, that is a 1-cell $s:A \to A$ in $\mathcal{K}$ and a 2-cell $\lambda : st \to ts$ satisfying axioms as above.
3. Two 2-cells $\sigma : 1 \to (s,\lambda)$ and $\mu : (s, \lambda)(s,\lambda) \to (s, \lambda)$ in $KL(\mathcal{K})$, which amount to two 2-cells in $\mathcal{K}$, $\sigma : 1 \to ts$ and $nu : ss \to ts$, satisfying axioms as above.

A morphism of cowreaths is a 1-cell $(f,\phi, \bar \phi) : (A,t,s) \to (A',t',s')$ in $KL(KL(\mathcal{K}))$. It consists of

1. A colax morphism of monads $(f,\phi):(A,t) \to (A',t')$, which is a 1-cell $f:A \to A'$ in $\mathcal{K}$ plus a comparison 2-cell $\phi : tf \to ft'$ (satisfying analogous laws to a lax morphism of monads)
2. A 2-cell $\bar \phi : (f,\phi)(s, \lambda) \to (s,\lambda)(f,\phi)$ in $KL(\mathcal{K})$, which amounts to a 2-cell ${\bar{\phi}} : fs \to t's'f$, suitably commuting with $\phi$ and $\lambda$:

The pair $((f,\phi), \bar \phi)$ is then required to satisfy the axioms of 1-cell in $KL$.

Related concepts

• distributive law

Literature

• 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.

• Dimitri Chikhladze, A note on warpings of monoidal structures, arXiv:1510.00483 (2015).