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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 $TM$.

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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 $T$.

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  • $\begingroup$ Hm. Thanks for the answer ~ Interesting. In both of your objections the problem seems to boil down to some kind of conjugacy with respect to an automorphism (I guess all such minimal finite subsets $M'$ have the same size). Regarding the "(-1)-truncation" example, my first intuition was to pick an arbitrary element of $M$ as the support of the single element of $TM$. I.e. $supp(\ast)=\{\ast\}$ for $M$ not empty. Maybe it's possible to modify/chose "the implied category of such data" in a way that avoids this obstruction. ... Something to think about. $\endgroup$
    – Gerrit Begher
    Commented May 14, 2021 at 14:06
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A similar notion (at least in spirit) was introduced by Simon Henry in An abstract elementary class non-axiomatizable in $L_{(\infty,\kappa)}$, see Def. 4.2 and later used by Michael Lieberman, Jiří Rosický, Sebastien Vasey in Hilbert spaces and $C^∗$-algebras are not finitely concrete.

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