On the iteration of weak wreath products (Q2884473)

\textit{S. Caenepeel} and \textit{E. de Groot} [Contemp. Math. 267, 31--54 (2000; Zbl 0978.16033)] generalized the entwining structures between an algebra and coalgebra to the weak context in the sense of the present author [Weak Hopf algebras and their application to spin models. Budapest (PhD Thesis) (1997)]. For a more general context see [\textit{J. N. Alonso Álvarez} et al., Bull. Aust. Math. Soc. 80, No. 2, 306--316 (2009; Zbl 1205.18006)] and the reviewer's [Theory Appl. Categ. 22, 313--320 (2009; Zbl 1201.18004)].NEWLINENEWLINEIn a much earlier work [J. Pure Appl. Algebra 2, 149--168 (1972; Zbl 0241.18003)], the reviewer pointed out that the construction of the 2-category \(\mathrm{Mnd}\mathcal{K}\) of monads in a 2-category \(\mathcal{K}\) defines a monad on the 3-category of 2-categories and that the objects of the iterated construction \(\mathrm{Mnd}^2\mathcal{K} = \mathrm{Mnd}\mathrm{Mnd}\mathcal{K}\) are distributive laws between two monads. The multiplication of the monad \(\mathrm{Mnd}\) takes a distributive law to the composite of the two monads. This composite can also be thought of as a wreath product.NEWLINENEWLINEFurther iteration of \(\mathrm{Mnd}\) was looked at by \textit{E. Cheng} [Math. Proc. Camb. Philos. Soc. 150, No. 3, 459--487 (2011; Zbl 1255.18004)]. The author defines a sequence of 2-categories \(\mathrm{Wdl}^{(n)}\mathcal{K}\) constructed from \(\mathcal{K}\), whose objects are \(n\)-fold weak distributive laws, with \(n\) weak wreath product 2-functors into \(\mathrm{Wdl}^{(n-1)}\mathcal{K}\) when idempotents split in \(\mathcal{K}\). An example of an iterated weak wreath product is provided in the algebra of observable quantities in an Ising-type quantum spin chain, where the spins take values in a dual pair of finite weak Hopf algebras. The \(n\)-ary weak wreath products are characterized in \(\mathcal{K}\) when idempotents split.