Correspondence between operads and monads requires tensor distribute over coproduct?

Question

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without assuming that the monoidal product distributes over coproducts. But no such assumption is mentioned in my sources (for example Operads, Algebras, Modules by May (PDF).) Am I missing something in my argument?

Let $\mathcal{C}$ (mathcal C) be an operad (for simplicity take it non-symmetric) over a symmetric monoidal category $\mathcal{V},$ with composition $\gamma\colon \mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes\dotsb\otimes\mathcal{C}(m_n)\to\mathcal{C}(m_1+\dotsb+m_n).$ We define a functor $C\colon \mathcal{V}\to\mathcal{V}$ (Roman C) by $CX = \coprod_i \mathcal{C}(i)\otimes X^{\otimes i}$.

Then one wants to verify that the operad structure $\gamma$ gives a monad on $C$. That is, we need a natural morphism $C^2X\to CX,$ or $\coprod_i \mathcal{C}(i)\otimes \left(\coprod_j \mathcal{C}(j)\otimes X^{\otimes j}\right)^{\otimes i}\to \coprod_k \mathcal{C}(k)\otimes X^{\otimes k}$. By universal prop of coproducts, it will suffice to exhibit an arrow $\mathcal{C}(n)\otimes \left(\coprod_j \mathcal{C}(j)\otimes X^{\otimes j}\right)^{\otimes n}\to \coprod_k \mathcal{C}(k)\otimes X^{\otimes k}$ for all $n$.

Clearly we have $\mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes X^{\otimes m_1}\otimes \dotsb \otimes \mathcal{C}(m_n)\otimes X^{\otimes m_n}\to \mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes\dotsb\otimes \mathcal{C}(m_n) X^{\otimes m_1+\dotsb+m_n}\to \\ \mathcal{C}(m_1+\dotsb+m_n)\otimes X^{\otimes m_1+\dotsb+m_n}\to\coprod_k\mathcal{C}(k)\otimes X^{\otimes k},$ where the first arrow is by symmetry of the monoidal structure, the second arrow is the operad composition $\gamma,$ and the third arrow is the canonical inclusion into the coproduct.

Therefore by universal prop of coproducts, we have $\coprod_{m_1,\dotsc,m_n}\mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes X^{\otimes m_1}\otimes \dotsb \otimes \mathcal{C}(m_n)\otimes X^{\otimes m_n}\to\coprod_k\mathcal{C}(k)\otimes X^{\otimes k}.$

In general, again using inclusion morphisms of coproducts, we have arrows $\mathcal{C}(m_\ell)\otimes X^{\otimes m_\ell}\to \coprod_j \mathcal{C}(j)\otimes X^{\otimes j}.$ Then by functorality of the monoidal product, we have $\mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes X^{\otimes m_1}\otimes \dotsb \otimes \mathcal{C}(m_n)\otimes X^{\otimes m_n}\to \mathcal{C}(n)\otimes \left(\coprod_j \mathcal{C}(j)\otimes X^{\otimes j}\right)^{\otimes n}$. By universal property of coproducts, we therefore have an arrow from the coproduct $\coprod_{m_1,\dotsc,m_n}\mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes X^{\otimes m_1}\otimes \dotsb \otimes \mathcal{C}(m_n)\otimes X^{\otimes m_n}\to \mathcal{C}(n)\otimes \left(\coprod_j \mathcal{C}(j)\otimes X^{\otimes j}\right)^{\otimes n}$.

To summarize, we have the obvious maps $\mathcal{C}(n)\otimes \left(\coprod_j \mathcal{C}(j)\otimes X^{\otimes j}\right)^{\otimes n}\leftarrow \coprod_{m_1,\dotsc,m_n}\mathcal{C}(n)\otimes \mathcal{C}(m_1)\otimes X^{\otimes m_1}\otimes \dotsb \otimes \mathcal{C}(m_n)\otimes X^{\otimes m_n} \to \\ \coprod_k \mathcal{C}(k)\otimes X^{\otimes k}.$ Unless we know that the arrow on the left is an isomorphism, we do not get the structure map for a monad on $C$. And that arrow on the left will generally not be an isomorphism if the monoidal product in $\mathcal{V}$ does not distribute over the coproduct. For example, if the monoidal product is the coproduct itself.

Answer 1

I also noticed this at some point. I think you are right. One reference where this assumption is explicitly spelled out is the paper of Getzler and Jones.

Answer 2

The correspondence between operads and monads still holds in the absence of commutation of coproducts and tensor products if one chooses correct definitions.

Operads are monoids in symmetric sequences equipped with the substitution product, algebras and modules over operads are (left) modules over monoids (concentrated in degree 0 in the case of algebras), and the associated monad is the functor O∘−, with the multiplication and unit maps coming from the monoid structure of O.

The substitution product defines a monoidal category in the usual sense only if the monoidal product preserves infinite coproducts. In the general case one obtains a monoidal category in which the associators need not be invertible. These are known as (op)lax (normal) monoidal categories and are defined as (op)lax monoids in the 2-category of categories equipped with the cartesian product. (Oplax/lax is chosen according to the direction in which the noninvertible associator goes.) See the paper “Lax monoids, pseudo-operads, and convolution” by Day and Street for more details.

The notions of monoids and left modules over monoids can be generalized to this setting, as explained in detail in Ching's paper “A note on the composition product of symmetric sequences”. If one also generalizes monads to this setting, one recovers the usual correspondence between operads and monads.

All of this becomes important in the case of cooperads, for which one would need tensor products to commute with infinite products to obtain the usual notion of a comonad from a cooperad. Although monoidal products often preserve infinite coproducts, they rarely preserve infinite products, unless the monoidal structure is cartesian, which forces one to consider the notions mentioned above.