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Started an article on monoidal monad. An earlier redirect had sent it over to Hopf monad which is something that Zoran was working on, but I think it deserves an article to itself, with discussion of the relation to commutative monads, etc. (which I have started).
Excellent, thanks!
Does anybody know of an example of a monoidal monad on a symmetric monoidal category, where the monad itself is not symmetric, i.e. the square
$\begin{matrix} TA \otimes TB &\to& T(A\otimes B)\\ \downarrow && \downarrow\\ TB \otimes TA &\to& T(B\otimes A)\\ \end{matrix}$doesn’t necessarily commute?
I thought that André Joyal mentioned such a monad at a CT talk about the Dold Kan correspondence and Tom Leinster blogged about it, but although I found the blogpost here, it doesn’t mention any such monad.
Does anybody happen to remember the monad in question, or another non-symmetric monoidal monad on a symmetric monoidal category?
Interesting question! Do cubical sets perhaps provide an example? I’d guess that they can be viewed as algebras for some monoidal monad on the category $\mathsf{Set}^{\mathbb{N}}$ (with symmetric monoidal structure given by levelwise product), or something like that (thinking of each element of $X(n)$, for some $n \in \mathbb{N}$, as a free-standing $n$-cube), with the monoidalness of the monad inducing the usual tensor product of cubical sets; but this tensor product is not symmetric, and I think this would necessitate that the square you mention does not always commute.
Hmm, I think the square can only be defined if the tensor product is symmetric (or at least braided) since the vertical maps are meant to be braiding maps. Anyway, which non-symmetric tensor product are you referring to? A kind of substitution tensor product?
EDIT: I assume that you mean the convolution tensor product on presheaves on this cube category. That’s indeed not symmetric.
Yes, that is the tensor product I meant: the one induced by Day convolution from the monoidal structure of the cube category.
Hmm, I thought that you were only using the symmetric structure of the underlying symmetric monoidal category to define that square, not the one on algebras? My feeling was that if that square commutes, it will force the monoidal structure on algebras to be symmetric, but I haven’t gone through the details.
One thing that I think is true is that the only way to get an example would be to have left and right strengths where the right strength is not compatible with the symmetric structure of the underlying symmetric monoidal category, but where one still has the necessary compatibility between the strengths to have a commutative monad (as defined in the non-symmetric setting). I think there should be such examples (I see no reason for the compatibility between strengths to force that the right strength comes from the symmetric structure), but if my idea about looking at non-symmetric-ness of the tensor product on algebras is no good, then I’m not sure where to look for them!
have tried to bring more logic into the section outline:
Now there is a section “Properties” with three subsections, the first of which is “Relation to commutative strong monads”.
The claims in this subsection probably have canonical references, but essentially none are given currently.
have added (here) the following proposition, with pointer to Kock (1972), Thm. 2.3:
For a monad $T$ on (the underlying category of) a symmetric closed monoidal category, there is a bijection between the structure on $T$ of a:
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