On the iteration of weak wreath products

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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 nite 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 sucfficient and necessary condition on a monad in K to be isomorphic to an n-ary weak wreath product.

Original languageEnglish
Pages (from-to)30-59
Number of pages30
JournalTheory and Applications of Categories
Publication statusPublished - Jan 30 2012


  • Monad
  • N-ary weak wreath product
  • Quantum spin chain
  • Weak distributive law
  • Yang-Baxter equation

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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