# On the iteration of weak wreath products

## Gabriella Böhm

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 $cal K$ and for any non-negative integer $n$, we introduce 2-categories $\Wdl^{(n)}(\cal K)$, of $(n+1)$-tuples of monads in $\cal K$ pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance $\Wdl^{(0)}(\cal K)$ coincides with $\Mnd(\cal K)$, the usual 2-category of monads in $\cal K$, and for other values of $n$, $\Wdl^{(n)}(\cal K)$ contains $\Mnd^{n+1}(\cK)$ as a full 2-subcategory. For the local idempotent closure $\overline \cal K$ of $\cal K$, extending the multiplication of the 2-monad $\Mnd$, we equip these 2-categories with $n$ possible weak wreath product' 2-functors $\Wdl^{(n)}(\ocK)\to \Wdl^{(n-1)}(\overline \cal K)$, such that all of their possible $n$-fold composites $\Wdl^{(n)}(\overline \cal K)\to \Wdl^{(0)}(\overline \cal K)$ are equal; that is, such that the weak wreath product is associative'. Whenever idempotent 2-cells in $\cal K$ split, this leads to pseudofunctors $\Wdl^{(n)}(\cal K)\to \Wdl^{(n-1)}(\cal K)$ 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 $\Wdl^{(n)}(\overline \cal K)$ into the 2-category of commutative $n+1$ dimensional cubes in $\Mnd(\overline \cal K)$ (hence into the 2-category of commutative $n+1$ dimensional cubes in $\cal K$ whenever $\cal K$ has Eilenberg-Moore objects and its idempotent 2-cells split). Finally we give a sufficient and necessary condition on a monad in $\overline \cal K$ to be isomorphic to an $n$-ary weak wreath product.

Keywords: monad, weak distributive law, n-ary weak wreath product, Yang-Baxter equation, quantum spin chain

2000 MSC: 18C15, 18D05, 16W30

Theory and Applications of Categories, Vol. 26, 2012, No. 2, pp 30-59.

Published 2012-01-30.

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