On the iteration of weak wreath products

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; i.e. 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.