# Functors from $\mathsf{Set}$ to $\mathsf{Mon}$?

There is the well-known functor which sends a set $X$ in $\mathsf{Set}$ to the free monoid on $X$ living in $\mathsf{Mon}$ (a functor with distinctly interesting properties, like being adjoint to the forget functor from monoids to sets).

There are the boring constant functors, where such a functor sends each and every set $X$ in $\mathsf{Set}$ to some chosen monoid $M$ and sends every set arrow to the identity morphism $1_M$ on that monoid (distinctly uninteresting!).

The existence of the second class makes the point that functoriality doesn't by itself fix what a functor from $\mathsf{Set}$ to $\mathsf{Mon}$ has to look like. But still, are there other interesting non-constant functors from $\mathsf{Set}$ to $\mathsf{Mon}$ which are worthy of note?

(And if a follow-up is allowed, is the situation e.g. with functors $\mathsf{Set}$ to $\mathsf{Grp}$ or $\mathsf{Set}$ to $\mathsf{Rng}$ interestingly different?)

• The free abelian monoid, the free abelian monoid of exponent $n$, the composition of any functor from set to something with a functor to Mon (the functor mapping a set $X$ to Brauer group of the field of rational functions $\mathbb C(X)$, say, to give an pretty absurd example). There are many! – Mariano Suárez-Álvarez Sep 28 '15 at 19:54
• Excellent -- just the sort of thing I wanted pointers to! – Peter Smith Sep 28 '15 at 19:58

There are tons and tons of such functors. Given any such functor you can compose it with either an endofunctor of $\text{Set}$ or an endofunctor of $\text{Mon}$, and there are tons of those. So let me first restrict attention to the nicest ones, which are the ones that preserve colimits.
$\text{Set}$ turns out to be the free cocomplete category on a point: that is, if $C$ is any cocomplete category, then the category of cocontinuous functors $F : \text{Set} \to C$ is equivalent to $C$ itself, with the equivalence given in one direction by evaluating $F \mapsto F(1)$, and in the other direction by sending an object $c \in C$ to the functor
$$X \mapsto \coprod_X c.$$
Setting $c = \mathbb{Z}_{\ge 0}$ to be the free monoid on a generator gives the free monoid functor, but other choices are possible. For example, setting $c = \mathbb{Z}$ gives the free group functor.
Now you can precompose with endofunctors $\text{Set} \to \text{Set}$, such as the functors $X \mapsto X^n$ or $X \mapsto {X \choose n}$ (this is notation for the set of $n$-element subsets of $X$). Or postcompose, or take coproducts, or... I mean, there's really just a lot of things you can do.