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(K), of (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl(K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl(K) contains Mnd(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(K) → Wdl(n−1)(K), such that all of their possible n-fold composites Wdl(K) → Wdl(K) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2cells in K split, this leads to pseudofunctors Wdl(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(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.