Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A monad $(X, A)$ in $C$ is defined as a monoid object $(A : X \rightarrow X, p^A : AA \rightarrow A, e^A: 1_X \rightarrow A)$ in $C(X, X)$. Given two monads $(X, A)$ and $(X, B)$, a distibutive law is a 2-cell $d : BA \rightarrow AB$ satisfying few axioms. One of the bottom lines is that, a distributive law gives rise to a monad structure on the composite $AB$. We also have the following characterization:
Theorem: There is a one-to-one correspondence between distributive laws between monads $(X, A)$ and $(X, B)$ and those monad structures on $AB$ for which $e^A1_B : B \rightarrow AB$ and $1_Ae^B : A \rightarrow AB$ are monoid morphisms and $p^{AB}(1_Ae^Be^A1_B) = 1_{AB}$.
Proof: From a distrubutive law one defines $p^{AB} = (p^Ap^B)(1_Ad1_B)$ and $e^{AB} = e^Ae^B$. Vice versa, from a monad structure on $AB$ one defines $d = (e^A1_{BA}e^B)$.
Suppose now that $(X, B)$ is a monad, but $A$ is only an endomorphism $X \rightarrow X$. Then we can ask a question:
Q: What is a structure corresponding to a monad structure on $AB$ compatible in a certain sense with the monad structure on B?
The following is an answer:
A: Consider a structure consisting of 2-cells $t : ABA \rightarrow AB$ and $k : 1_X \rightarrow BA$ satisfying:
- $(1_Ap^B)(t1_B)(1_{AB}t) = p(Ap^BA)(tBA)$
- $(Ap^B)(tB)(kAB) = 1_{AB}$
- $t(kA) = Ae^B$.
Then, we have a one-to-one correspondence: Given $t$ and $k$ we define $p^{AB} = (1_Ap^B)(t1_B)$ and $e^{AB} = k$. Visa versa, from a (good enough) monad structure on $AB$ we define $t = p^{AB}(1_{ABA}e^B)$ and $k = e^{AB}$.
My question is: Does anyone know an interesting example of such a structure?
In particular, does something like this come up interestingly in the context of operads? (Thinking of operads as monads). In this situation, from an operad $B_n$, a collection $A_n$ and an extra structure we get a new operad. I guess if we think of $B_n$-s as objects of operations, we can think of $A_n$-s as objects of parameters which can not be composed on their own.