# What does the multiplication operation of a monad mean?

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.

• It means union. Jul 23, 2019 at 17:30
• Thanks. Can you be more specific?
– Tim
Jul 23, 2019 at 18:47

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 $$\mu([[a,a,a],[],[a,b,b,c],[c,a,b]]) = [a,a,a,a,b,b,c,c,a,b]$$.