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?

]]>As far as I can tell, the difference between a monadic functor and a strictly monadic functor boils down to this: a strictly monadic functor $U : \mathcal{D} \to \mathcal{C}$ has the property that, for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a *unique* object $\tilde{C}$ and an (automatically unique) isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$. Conversely, any monadic functor with this property is strictly monadic. This is very reminiscent of the definition of isofibration, but a monadic isofibration need not be strictly monadic. What’s a good phrase to describe functors like these for which isomorphisms lift uniquely?

- “Creates isomorphisms” on its own might be construed as “conservative”.
- “Isomorphisms lift uniquely” suggests something a little bit stronger than what I’m going for – the unique functor from a group to the terminal category admits unique solutions to the object part of the lifting problem (for obvious reasons) but not the isomorphism part.
- Maybe “strong/strict isofibration”…?

The Eilenberg–Moore construction is essentially a 2-functor from the “2-category of monads on locally small categories” to the “2-category of locally small categories”, modulo certain size issues. In the reverse direction, it is known that any functor $\Phi : \mathcal{C}^\mathbb{S} \to \mathcal{D}^\mathbb{T}$ such that $U^\mathbb{T} \Phi = F U^\mathbb{S}$ for some functor $F : \mathcal{C} \to \mathcal{D}$, where $U^\mathbb{S} : \mathcal{C}^\mathbb{S} \to \mathcal{C}$ and $U^\mathbb{T} : \mathcal{D}^\mathbb{T} \to \mathcal{D}$ are the respective forgetful functors, must come from a unique morphism of monads $\mathbb{S} \to \mathbb{T}$. (This is basically a stronger version of Theorem 6.3 in *Toposes, triples and theories*.) It is also not hard to come up with functors $\mathcal{C}^\mathbb{S} \to \mathcal{D}^\mathbb{T}$ that do *not* arise in this fashion.

But what about the 2-cells? Is every natural transformation between functors induced from a morphism of monads also induced by a 2-cell between the monad morphisms? Clearly, the answer is no – any natural transformation that doesn’t factor through the forgetful functor $U^\mathbb{S}$ can’t be induced by a 2-cell between monad morphisms. But I haven’t been able to come up with a proof or counterexample when I assume that the natural transformation *does* factor through $U^\mathbb{S}$.