Consider the *word* a.k.a. *list monad* $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:


$$
\mathrm{supp}(w):=\{\textrm{ all elements of $M$ occurring in $w$ }\} \subseteq M
$$


that we could call the ***support of $w$***.


**Question:** Is there something similar for arbitrary finitary endofunctors $T:\mathrm{Set}\to\mathrm{Set}$?


What I imagine to find is a datum that looks something like this:


* *Support:* A map $\mathrm{supp}\_T:TM\to\mathrm{FinSet}/M$ assigning to every element $w\in TM$ a finite set over $M$:
$$\pi\_w:\mathrm{supp}\_T(w)\to M$$
* *Representatives:* A map $\mathrm{rep}\_T:TM\to\coprod\_{w\in TM}T(\mathrm{supp}\_T(w))$ such that $\mathrm{rep}\_T(w)\in T(\mathrm{supp}\_T(w))$ and
$$T(\pi\_w)(\mathrm{rep\_T}w)=w$$
* *Minimality:* This datum should also be initial(?) in the implied category of such data.


**Idea:** The intuition is that since $T$ is finitary, the elements of $TM$ can be defined in a "finite way"; We don't need the whole of $M$ to describe a single element of $TM$.



Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary
This seems like too much to hope for at this level of generality. For example take $T$ to be the "(-1)-truncation" functor
$$
T(\varnothing) = \varnothing, \qquad T(M) = \* \,\, \mbox{for $M \ne \varnothing$}.
$$
Then let $M$ be an arbitrary infinite set. The support map would have to take the element of $TM$ to a choice of a finite set with a map to $M$, and the finite set has to be nonempty. But there is no way to choose this data compatibly with the automorphisms of $M$.


What you do have in general is that, since any set $M$ is the filtered colimit of its finite subsets (along the subset inclusions) and $T$ is finitary, any element of $TM$ arises as the image of an element of $TM'$ for *some* finite subset $M' \subset M$, and then also for any finite subset bigger than $M'$. But trying to pick out a distinguished $M'$ (say as the smallest one) won't succeed unless you have some extra information about $T$.


