nLab
wreath
Wreaths
Context
Higher algebra
2-Category theory
2-category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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 K K 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 EM EM denotes the completion of a 2-category under Eilenberg–Moore objects . Since EM EM is a 2-monad , the multiplication EM ∘ EM → EM EM \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:
A monad ( A , t , η , μ ) (A,t, \eta, \mu) in 𝒦 \mathcal{K} ,
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 : A → A s:A \to A in 𝒦 \mathcal{K} and a 2-cell λ : ts → st \lambda : ts \to st satisfying the axioms:
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 : 1 → st \sigma : 1 \to st and nu : ss → st nu : 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 ) op KL(\mathcal{K}) = EM(\mathcal{K}^{op})^{op} is the free Kleisli completion of 𝒦 \mathcal{K} .
It consists of
A monad ( A , t , η , μ ) (A,t, \eta, \mu) in 𝒦 \mathcal{K} ,
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 : A → A s:A \to A in 𝒦 \mathcal{K} and a 2-cell λ : st → ts \lambda : st \to ts satisfying axioms as above.
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 : 1 → ts \sigma : 1 \to ts and nu : ss → ts nu : 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
A colax morphism of monads ( f , ϕ ) : ( A , t ) → ( A ′ , t ′ ) (f,\phi):(A,t) \to (A',t') , which is a 1-cell f : A → A ′ f:A \to A' in 𝒦 \mathcal{K} plus a comparison 2-cell ϕ : tf → ft ′ \phi : tf \to ft' (satisfying analogous laws to a lax morphism of monads )
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 ¯ : fs → t ′ s ′ f {\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 KL KL .
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.
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