Is there a canonical product on the category of monads on Set?

Question

I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists when there is a distributive law. I want to be able to do familiar monoidal moves, which I will explain. Given these monads, $M_S = \langle S , \eta_S, \mu_S \rangle$ and $M_T = \langle T , \eta_T, \mu_T \rangle$, suppose you have a distributive law, $\lambda: ST \rightarrow TS$. The underlying endofunctor of the composite is $ST$. Now say you have some monad maps $F_L: M_S \rightarrow Q$ and $F_R: M_T \rightarrow Q'$. I want a monoidal product, where I can write $M_S \otimes M_T$, and so that I can do the following:

$$F_L \otimes I : M_S \otimes M_T \rightarrow Q \otimes M_T$$ $$I \otimes F_R : M_S \otimes M_T \rightarrow M_S \otimes Q'$$

I guess the problem is that there can be more that one distributive law given two monads. The fact that there can be zero distributive law is handled by this product being partial. But the other way around means that monad composition cannot be subsumed in a product. Has anyone seen any monoidal products defined on the category of monads on Set?

There is some language I am seeing but I don't understand: "composite along a distributive law" and "connection between the distributive law and $\otimes$". People seem to be implying that if I give a"connection" between the distributive law and the tensor, this might be an intelligible question. How do I give a connection between the distributive law tensor product?

Answer 1

I don't know of any monoidal product in general, but there is an equivalence between distributive laws and multiplications on their composition which produce monoids when combined with the obvious unit map.

One direction is clear: given a distributive law $\lambda:TS\to ST$, we can form the map $$STST\to SSTT\to ST\,,$$ and this will serve as a multiplication for the monoid. The associativity of this multiplication will follow from a suitable assumption on the distributive law (see the full definition here: https://ncatlab.org/nlab/show/distributive+law#:~:text=%E2%88%98lM).-,A%20distributive%20law%20from%20a%20monad%20T%3D(T%2C%CE%BC,(l)%E2%88%98lP. ).

To go the other direction, assume that there is a multiplication $\mu_{ST}:STST\to ST$. Then the composition $$TS\mathop{\longrightarrow}\limits^{\eta_S\circ\text{id}\circ\eta_T}STST\to ST$$ can be shown to be a distributive law. The required properties follow from the assumption of associativity of $\mu_{ST}$. This can either be seen by liberally pasting together commutative squares, or equivalently by using a string diagram argument.

In order for this to be compatible with your map $F_L:S\to Q$, one sufficient condition would be if you assume that $F_L$ is an epimorphism. This was mentioned in a comment on the previous version of this post, so my apologies are due to this commenter since I have forgotten their @. From here, the corresponding statement is also valid for $F_R$ and $Q'$ by symmetry.