1
$\begingroup$

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?)

$\endgroup$
  • 3
    $\begingroup$ 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! $\endgroup$ – Mariano Suárez-Álvarez Sep 28 '15 at 19:54
  • $\begingroup$ Excellent -- just the sort of thing I wanted pointers to! $\endgroup$ – Peter Smith Sep 28 '15 at 19:58
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.