Distributive laws between two monads in a 2-category $\CK$, as defined by JonBeck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to bemonads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the authordefined wreaths to be monads in a 2-category $\mathrm{EM}\CK$ of monads withdifferent 2-cells from $\mathrm{Mnd}\CK$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and,later, various others, they are comonads in $\mathrm{Mnd}\CK$. Actually, aspointed out by John Power and Hiroshi Watanabe, there are a number of dualpossibilities for mixed distributive laws. It is natural then to consider mixed wreaths as we do in this article, theyare comonads in $\mathrm{EM}\CK$. There are also mixed opwreaths: comonoids inthe Kleisli construction completion $\mathrm{Kl}\CK$ of $\CK$. The main examplestudied here arises from a twisted coaction of a bimonoid on a monoid.Corresponding to the wreath product on the mixed side is wreath convolution,which is composition in a Kleisli-like construction. Walter Moreira'sHeisenberg product of linear endomorphisms on a Hopf algebra, is an example ofsuch convolution, actually involving merely a mixed distributive law.Monoidality of the Kleisli-like construction is also discussed.
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机译：作者指出JonBeck在$ \ CK = \ mathrm {Cat} $情况下定义的2类$ \ CK $中两个单子之间的分布规律。作者指出是2类$ \ mathrm { MND} \ CK $。史蒂夫·拉克（Steve Lack）和作者将花环定义为2个类别的\\ mathrm {EM} \ CK $中的单子，其中2个单元格与$ \ mathrm {Mnd} \ CK $不同。乔恩·贝克（Jon Beck），迈克·巴尔（Mike Barr）等人也考虑了混合分配法则，它们在$ \ mathrm {Mnd} \ CK $中是同名的。实际上，约翰·鲍尔（John Power）和渡边博（Hiroshi Watanabe）指出，混合分配法存在多种双重可能性。然后自然要像我们在本文中那样考虑混合花圈，它们是$ \ mathrm {EM} \ CK $中的组合。也有不同的花招：Kleisli施工完成中的\\ mathrm {Kl} \ CK $ / $ \ CK $中的类彗星。这里研究的主要例子来自于类人动物在类半体上的扭曲作用。与混合侧上的花环积相对应的是花环卷积，它是一个类似于Kleisli的构造。沃尔特·莫雷拉（Walter Moreira）在Hopf代数上线性内同态的Heisenberg乘积是这种卷积的一个例子，实际上只涉及混合分布定律。还讨论了Kleisli式构造的单态性。
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