Monad Bimodule

Let $S:\mathcal{A}\to\mathcal{A}$, $T:\mathcal{B}\to\mathcal{B}$ be a pair of Monads. An S-T-bimodule consists of

• A Functor $F:\mathcal{A}\to\mathcal{B}$
• A Natural Transformation $\lambda:T\circ F\to F$
• A Natural Transformation $\rho:F\circ S\to F$

such that the following diagrams commute:

Properties

• Every Monad Algebra $A:\mathcal{C},\alpha:T(A)\to A$ is a $(mathrm{Id}{ifx..lse\par i},T)$-bimodule, where $mathrm{Id}{ifx..lse\par i}$ is taken as a Monad on the Terminal Category.

  The object $A:\mathcal{C}$ is a single-object diagram in $\mathcal{C}$, so our underlying 1-cell is simply $A:1\to\mathcal{C}$. Moreover, the right action is completely trivial, and the compatibility condition reduces to naturality of $\alpha$.

• Monad bimodules and Ring Bimodules are both instances of Bimodules in a Bicategory, so a lot of the theory of Ring Bimodules can be ported over to monad bimodules.