What does the multiplication operation of a monad mean?

Question

In the free monoid monad $(T, \eta, \mu)$ in category $Set$:

$T: Set\to Set$ is a endofunctor, is the composition of the free monoid functor $List:Set→Mon$ and the forgetful functor $U:Mon→Set$. $TA$ is the set of finite lists of elements in set $A$.

$\eta_A: A\to TA$ sends each element $a$ of set $A$ to the corresponding singleton list $[a]$.

What does the multiplication operation $\mu$ of a monad mean? Specifically, what does $\mu_A: T^2A \to TA$ do/mean?

Thanks.

Answer 1

The set $T(T(X))$ is the set of lists of lists and the natural transformation $\mu$ is just merging the lists of lists into one single list, also known as concatenation.

To be more specific, an element of $T(T(X))$ is something like $L = [[x_{1,1},x_{1,2},\cdots,x_{1,n_1}],\cdots, [x_{k,1},\cdots,x_{k,n_k}]]$, with $k,n_1,\cdots,n_k \in \mathbb{N}$, (when $n_i$<1 lets say the list is empty). Then $\mu(L) = [x_{1,1},\cdots,x_{k,n_k}]$.

To give an example lets say $X = \{a,b,c\}$, then $\mu([[a,a,a],[],[a,b,b,c],[c,a,b]]) = [a,a,a,a,b,b,c,c,a,b]$.