Monad

Remarks

Example: the power set monad on the category of sets. Here $T(X)$ is the power set of $X$, and $T(f)$ sends a subset $A$ to its image $f(A)$. The unit is $\eta_X(x) = \set{x}$ and the multiplication is $\mu_X(\mathcal{S}) = \bigcup \mathcal{S}$. Both sides of the associativity law send a set of sets of subsets to the union of all the subsets that occur in it. The unit laws hold because $\bigcup \set{A} = A$ and $\bigcup \set{\set{a} : a \in A} = A$.

Non-example: on the category of sets let $T(X) = X \times X$, $\eta_X(x) = (x, x)$ and $\mu_X((a, b), (c, d)) = (a, c)$. Then $(\mu \circ \eta T)_X$ sends $(x, y)$ to $\mu_X((x, y), (x, y)) = (x, x)$. This differs from $(x, y)$ when $x \ne y$, so a unit law fails. With $\mu_X((a, b), (c, d)) = (a, d)$ instead, one gets a monad.

A monad on $\mathcal{C}$ is the same as a monoid object in the monoidal category of endofunctors of $\mathcal{C}$, with composition as the product. The dual notion is a comonad.