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It is well known that any operad on a nice monoidal category induces a monad.

I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the induced monad $\hat T$ on $\cal V$ commutes with finite co/limits.

Thanks a lot!

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  • $\begingroup$ Preservation of terminal objects seems like a tall order, at least when the monoidal category is cartesian. If the operad is called $P$ then this would give $\sum_n P_n = \hat{T}(1) = 1$. But maybe you're not interested in the cartesian case. If not, could you give a hint as to what kind of example you are interested in? $\endgroup$ Oct 22, 2014 at 16:26
  • $\begingroup$ I hoped in your answer! I'm not interested in some particular example: I agree that in the cartesian case the condition is rather strong and unnatural: I'm basically interested in cases where the monoidal structure arises as a "tensor between algebras" (modules over a ring? chain complexes?); I've been deliberately vague, but a more precise question should be: does any "famous" operad induce a cocontinuous monad? $\endgroup$
    – fosco
    Oct 22, 2014 at 17:32
  • $\begingroup$ I would expect this to almost never happen. A decategorified version of your question would be: which power series $f(x) = \sum_{n=0}^\infty \frac{a_n x^n}{n!}$ commute with sums, $f(x+y) = f(x) + f(y)$, which I think only happens (with characteristic 0 coefficients) if $a_n = 0$ for $n \neq 1$. This suggests that maybe you can find mild hypothesis on $\mathcal{V}$ so that $\hat{T}$ commutes with coproducts only if the operad is concentrated in degree 1. $\endgroup$ Oct 22, 2014 at 17:35
  • $\begingroup$ Also, you said "operad" but the question is only about the underlying symmetric sequence because you are interested in a property of just the functor part of the monad. $\endgroup$ Oct 22, 2014 at 17:36
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    $\begingroup$ Still commenting rather than answering because I haven't got anything nontrivial to say... but if you view a monoid $M$ as an operad (with no operations of arity $\neq 1$) then the resulting monad is $M \otimes -$, which preserves colimits if your monoidal category is closed. But that's all I can think of. $\endgroup$ Oct 22, 2014 at 17:49

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