Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl (n) (K), of (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl (0) (K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl (n) (K) contains Mnd n+1 (K) as a full 2-subcategory. For the local idempotent closure K of K, extending the multiplication of the 2-monad Mnd, we equip these 2-categories with n possible ‘weak wreath product ’ 2-functors Wdl (n) (K) → Wdl (n−1) (K), such that all of their possible n-fold composites Wdl (n) (K) → Wdl (0) (K) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2-cells in K split, this leads to pseudofunctors Wdl (n) (K) → Wdl (n−1) (K) obeying the associativity property up-to isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of Wdl (n) (K) into the 2-category of commutative n + 1 dimensional cubes in Mnd(K) (hence into the 2-category of commutative n + 1 dimensional cubes in K whenever K has Eilenberg-Moore objects and its idempotent 2-cells split). Finally we give a sufficient and necessary condition on a monad in K to be isomorphic to an n-ary weak wreath product.

We note an inversion property of the fusion map associated to many semibialgebras. Let C = (C, ⊗, I, c) be a symmetric (or just braided) monoidal category. A Von Neumann “core ” in C is firstly a semibialgebra in C, that is, an object A in C with an associative multiplication: µ: A ⊗ A − → A (µ3 = µ(1⊗µ) = µ(µ⊗1) : A⊗A⊗A − → A) and a coassociative comultiplication: δ: A − → A ⊗ A (δ3 = (1 ⊗ δ)δ = (δ ⊗ 1)δ: A − → A ⊗ A ⊗ A) such that: δµ = (µ ⊗ µ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A − → A ⊗ A It is also equipped with an endomorphism in C such that: S: A − → A µ3(1 ⊗ S ⊗ 1)δ3 = 1: A − → A The name “Von Neumann core ” stems partly from the notion of a Von Neumann regular semigroup, which is then a VN-core in Set, while the free vector space on it is a particular type of VN-core in Vect, and partly from the properties of the paths which generate a (row-finite) graph algebra[5]. The fusion map f = (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A − → A ⊗ A then satisfies the fusion equation by the semibialgebra axiom of A (see [6]), and if we set: g = (1 ⊗ µ)(1 ⊗ S ⊗ 1)(δ ⊗ 1) as a tentative “inverse ” to f, then we get the following (partial) results:

We discuss an inversion property of the fusion map associated to many semibialgebras. Please note that a characterisation of Hopf k-algebras has been added at the end of this version. Let C = (C, ⊗, I, c) be a symmetric (or just braided) monoidal category. A Von Neumann “core ” in C is firstly a semibialgebra in C, that is, an object A in C with an associative multiplication: µ: A ⊗ A − → A (µ3 = µ(1⊗µ) = µ(µ⊗1) : A⊗A⊗A − → A) and a coassociative comultiplication: δ: A − → A ⊗ A (δ3 = (1 ⊗ δ)δ = (δ ⊗ 1)δ: A − → A ⊗ A ⊗ A) such that: δµ = (µ ⊗ µ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A − → A ⊗ A It is also equipped with an endomorphism S: A − → A in C such that: µ3(1 ⊗ S ⊗ 1)δ3 = 1: A − → A The name “Von Neumann core ” stems partly from the notion of a Von Neumann regular semigroup, which is then precisely a VN-core in Set, while the free vector space on it is a particular type of VN-core in Vect, and partly from the properties of the paths which generate a (row-finite) graph algebra[5]. The fusion map f = (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A − → A ⊗ A