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I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is not necessarily monoidal. (Imagine left-tensoring with an algebra $A$ internal to $\mathcal{C}$, so $T = A \otimes -$.)

Is there a general way to decide whether the Kleisli category or the Eilenberg-Moore category are braided monoidal?

I already know from Wikipedia that if $T$ is lax monoidal and $\mu$ and $\eta$ are monoidal natural transformations, then the Kleisli category is monoidal as well, but what about the Eilenberg-Moore category? And do either have a braiding coming naturally from $\mathcal{C}$?

Finally, when are Kleisli and Eilenberg-Moore categories semisimple? For my example, I guess the Kleisli category is not semisimple, but the Eilenberg-Moore category is, but I'm still a bit confused.

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    $\begingroup$ "what about the Eilenberg-Moore category? And do either have a braiding coming naturally from $\cal C$?" Are you aware of this paper? arxiv.org/abs/1003.1920 $\endgroup$ – Fosco May 14 '14 at 16:28
  • $\begingroup$ @tetrapharmakon Gosh... I was at Alain's talk about this and didn't understand the connection to my question. So I was aware of this paper, but not of the relevance to the question. Is my example a Hopf monad? $\endgroup$ – Manuel Bärenz May 14 '14 at 16:33
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tetrapharmakon's hint is excellent. The article he refers to (and the earlier article http://arxiv.org/abs/math/0604180 by some of the same authors) define "quasi-triangular Hopf monads" that are an abstraction of quasi-triangular Hopf algebras. For these monads, the Eilenberg-Moore category is indeed braided. (I didn't find that paper probably because they never mention "Eilenberg-Moore category" but only refer to "the category of $T$-modules".)

Even better, my example $A \otimes -$ shows up there. This monad is quasi-triangular as a monad if $A$ is quasi-triangular as a Hopf algebra (a quantum group).

They also define similar conditions for semisimplicity, sovereign structures, ribbon structures.

In that article, another article by Moerdijk is mentioned (http://www.sciencedirect.com/science/article/pii/S0001870807001430), and it seems that he established that comonoidal monads (he calls them "Hopf monads", Alain Bruguières calls them "bimonads") have a monoidal Eilenberg-Moore category.

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