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This question on math.stackexchange piqued my interest. It asks what the monads on a given monoid are. That is, if we treat a monoid as a category with one object we can ask what the monads on that category are. I would have thought that this question would have had an elegant answer, because monads and monoids are both so fundamental. But in fact I can’t find a nice characterisation.
Writing down the definitions we find that a monad on a monoid $X$ is equivalent to “an endomorphism $\theta\colon X\to X$ together with two elements $m,h\in X$ such that:
$\forall x\in X$, $\theta(x)m = m\theta(\theta(x))$,
$\forall x\in X$, $\theta(x)h = hx$,
$m^2 = m\theta(m)$,
$m\theta(h) = mh = 1$.”
Now, in the nice cases when $X$ is a group or commutative, one can prove that $mh=hm=1$ and that $\theta$ is just the inner automorphism given by conjugation by $h$. But in the general case I’m not able to prove that $\theta$ can still only be an inner automorphism. So does anyone know what kind of structure this is?
I think it’s not too hard to come up with examples of monads on (the deloopings of) monoids that are not much simpler than monads on arbitrary categories. Suppose for instance that $\kappa$ is a cardinal number and $T$ a monad on $Set$ that preserves sets of cardinality $\kappa$; then $T$ restricts to a monad on the full subcategory of sets of cardinality $\kappa$, which has only one isomorphism class and hence is equivalent to the delooping of a monoid. For instance, if $\kappa=\aleph_0$ then $T$ could be any finitary algebraic theory (groups, monoids, rings, etc.).
I wrote some blog posts about Monads on Monoids and Adjunctions between Monoids. The most interesting thing I found was that each monad on a monoid arises from a unique adjunction between monoids. The Eilenberg-Moore construction either doesn’t give you a one-object category, or it’s equivalent to the Kleisli category!
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