nLab wreath

Wreaths

Context

Higher algebra

2-Category theory

Wreaths

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 KK can be seen as an object of Mnd(Mnd(K))Mnd(Mnd(K)), a wreath can be seen as an object of EM(EM(K))EM(EM(K)), where EMEM denotes the completion of a 2-category under Eilenberg–Moore objects. Since EMEM is a 2-monad, the multiplication EMEMEMEM \circ EM \to EM produces from every wreath a composite monad, called its wreath product.

Definition

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

  1. A monad (A,t,η,μ)(A,t, \eta, \mu) in 𝒦\mathcal{K},
  2. A 1-cell (s,λ):(A,t)(A,t)(s,\lambda):(A,t) \to (A,t) in EM(𝒦)EM(\mathcal{K}), that is a 1-cell s:AAs:A \to A in 𝒦\mathcal{K} and a 2-cell λ:tsst\lambda : ts \to st satisfying the axioms:
  1. Two 2-cells σ:1(s,λ)\sigma : 1 \to (s,\lambda) and μ:(s,λ)(s,λ)(s,λ)\mu : (s, \lambda)(s,\lambda) \to (s, \lambda) in EM(𝒦)EM(\mathcal{K}), which amount to two 2-cells in 𝒦\mathcal{K}, σ:1st\sigma : 1 \to st and nu:ssstnu : ss \to st, such that

Duality

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

It consists of

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

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

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

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

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).

Last revised on July 18, 2024 at 07:56:33. See the history of this page for a list of all contributions to it.