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?