In an earlier post, What is known about the category of monads on Set? the following observation was made:

What's more, all but two monads on Set have the property that there exists an algebra with more than one element. One of the exceptions is the monad M with M(A)=1 for all sets A; it's the theory generated by a single constant e and the equation x=e. The other is the monad M with M(A)=1 for all nonempty sets A and M(0)=0; that's the theory generated by no operations and the equation x=y.

Does anyone know a proof (or a reference to a proof) of this fact--that these are the only two monads on sets having trivial algebras?


I don't know where this observation was first made, but the proof is short.

Let $M$ be a monad on $Set$ such that every $M$-algebra has at most one element. For every set $A$, the set $M(A)$ has the structure of an $M$-algebra (a free one), so $M(A)$ has at most one element. On the other hand, the unit of the monad gives us a map $A \to M(A)$. Since there is no map from a nonempty set to the empty set, this implies that $M(A) = 1$ whenever $A$ is nonempty.

So, $M(\emptyset)$ is either $\emptyset$ or $1$, and $M(A) = 1$ for all nonempty $A$. In either case, $M$ becomes a monad in a unique way. This gives the two monads mentioned.


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