## Franklin's Notes

A monad is defined as a triple $\langle T,\eta,\mu\rangle$ where $T$ is an endofunctor of some category $\mathsf{X}$, and $\eta:1_\mathsf{X}\Rightarrow T$ and $\mu:T^2\Rightarrow T$ are natural transformations , which are required to be such that the following two diagrams commute:  The natural transformation $\eta$ is called the unit of the monad, and the natural transformation $\mu$ is called the multiplication of the monad. The commutativity of the first diagram is the associative law of the monad, and the second diagram expresses the left and right unit laws of the monad.

But what the heck does all of this this mean? These conditions are super obscure, so let's try and get some intuition about how to interpret what a monad actually does.

Right now, the way of conceptualizing a monad that's been most helpful for me has been to think of a monad as a kind of "wrapper" for data. The natural transformation $\eta:1_\mathsf{X}\to T$ expresses that any object in the category $\mathsf{X}$ can be wrapped in the kind of wrapper that $T$ represents, or that there is another object in the same category that is a "wrapped version" of the original object. The natural transformation $\mu:T^2\to T$, then, expresses that "double-wrapping" an object is in some sense redundant, and a double-wrapped version of an object can be reduced to a singly-wrapped versions of the same object.

Okay then, but under this interpretation, what do the two commutative diagrams mean?

The first commutative diagram addresses how a triply-wrapped object can be reduced to a singly-wrapped object. In particular, there are two potentially differend ways of reducing a triple-wrapping to a single-wrapping using $\mu$: we can either reduce the two outer wrappers to a single wrapper, and then reduce the combination of the resulting wrapper and the innermost wrapper; or we can reduce the two inner wrappers to a single wrapper, and then reduce the combination of the outermost wrapper and this newly formed inner wrapper to a single wrapper. The commutativity of this diagram expresses that these two ways of reducing a triple-wrapper are identical. The second commutativity condition, on the other hand, tells us about what happens if we add a second wrapper to pre-wrapped data and then immediately reduce the two wrappers to a single wrapper. In particular, adding another wrapper outside of the original wrapper and then reducing has the same effect as adding another wrapper inside of the original wrapper and then reducing. Every adjunction between functors gives rise to a monad. To be specific, if $F\dashv G$ with $F:\mathsf{C}\leftrightarrows\mathsf{D}:G$, then the endofunctor $GF:\mathsf{C}\to\mathsf{C}$ is a monad with unit given by $\eta:1_\mathsf{C}\Rightarrow T$, which is the same as the unit of the adjunction, and multiplication given by $\mu=G\epsilon F:T^2\Rightarrow T$, where $\epsilon$ is the counit of the adjunction.