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On the iteration of weak wreath products

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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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