TITLE: Examples of a monad in a monoid (i.e. category with one object)?
QUESTION [13 upvotes]: I've been trying to figure out what having a monad in a monoid (i.e. a category with one object) would mean.
As far as I can tell it would be a homomorphism (functor) $T : M → M$, with two elements (natural transformation components) $\eta, \mu : M$, such that

$\forall x. \eta x = T(x) \eta$
$\forall x. \mu T(T(x)) = T(x) \mu$
$\mu \eta$ = $\mu T(\eta)$ = 1
$\mu T(\mu) = \mu \mu$

The identity monad $T(x) = x$, with $\eta = \mu = 1$, is an obvious example for any monoid. But no other examples really come to mind... These laws seem a bit strange. Are there any interesting examples, or any good intuition for what the laws would mean?

REPLY [2 votes]: I wrote some blog posts about Monads on Monoids and Adjunctions between Monoids. The main results are:

If $M$ is commutative, a group, or finite, then every monad on it is a conjugation.
Every monad on monoids arises from a unique adjunction between monoids.
There are examples of monads on monoids that aren't given by conjugation. These are necessarily quite complicated, since by the first bullet point they must be infinite noncommutative monoids that aren't groups. One example is given by taking $M$ to be the monoid of order preserving functions $\mathbb N \to\mathbb N$ which are eventually translations, i.e. with the property that for all sufficiently large $n\in\mathbb N$ they have the form $n\mapsto n+c$ for some $c\in\mathbb Z$. We then define $\eta$ by $\eta(n)=n+1$, $\mu$ by $\mu(n+1)=n$ and $\mu(0)=0$, and $T$ by $T(f)(n+1) = f(n)+1$ and $T(f)(0)=0$.