← Levels and Natural Transformations of the Functor F_A

FirstVersion.tex

\begin{filecontents*}{fa_refs.bib}
@article{adamek-milius-moss-urbat2015,
  author       = {Ji{\v r}{\'\i} Ad{\'a}mek and Stefan Milius and Lawrence S. Moss and Henning Urbat},
  title        = {On finitary functors and their presentations},
  journal      = {Journal of Computer and System Sciences},
  volume       = {81},
  number       = {5},
  pages        = {813--833},
  year         = {2015},
  doi          = {10.1016/j.jcss.2014.12.002}
}

@misc{dahlqvist-neves2018,
  author       = {Fredrik Dahlqvist and Renato Neves},
  title        = {Compositional semantics for new paradigms: probabilistic, hybrid and beyond},
  year         = {2018},
  eprint       = {1804.04145},
  archiveprefix= {arXiv},
  primaryclass = {cs.LO}
}

@article{gumm-schroeder2001,
  author       = {H. Peter Gumm and Tobias Schr{\"o}der},
  title        = {Monoid-labeled transition systems},
  journal      = {Electronic Notes in Theoretical Computer Science},
  volume       = {44},
  number       = {1},
  pages        = {185--204},
  year         = {2001},
  doi          = {10.1016/S1571-0661(04)80908-3}
}

@misc{hora-kamio-maehara2025,
  author       = {Ryuya Hora and Yuhi Kamio and Yuki Maehara},
  title        = {Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets},
  year         = {2025},
  eprint       = {2503.03439},
  archiveprefix= {arXiv},
  primaryclass = {math.CT}
}

@article{kelly-lawvere1989,
  author       = {G. M. Kelly and F. W. Lawvere},
  title        = {On the complete lattice of essential localizations},
  journal      = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de Belgique. S{\'e}rie A},
  volume       = {41},
  number       = {2},
  pages        = {289--319},
  year         = {1989}
}

@article{kennett-riehl-roy-zaks2011,
  author       = {Carolyn Kennett and Emily Riehl and Michael Roy and Michael Zaks},
  title        = {Levels in the toposes of simplicial sets and cubical sets},
  journal      = {Journal of Pure and Applied Algebra},
  volume       = {215},
  number       = {5},
  pages        = {949--961},
  year         = {2011},
  doi          = {10.1016/j.jpaa.2010.07.002}
}

@misc{kori-watanabe2025,
  author       = {Mayuko Kori and Kazuki Watanabe},
  title        = {A No-go Theorem for Coalgebraic Product Construction},
  year         = {2025},
  eprint       = {2504.06592},
  archiveprefix= {arXiv},
  primaryclass = {cs.LO},
  note         = {To appear in FoSSaCS 2026}
}

@article{menni2019,
  author       = {Mat{\'\i}as Menni},
  title        = {Monic skeleta, boundaries, {A}ufhebung, and the meaning of `one-dimensionality'},
  journal      = {Theory and Applications of Categories},
  volume       = {34},
  number       = {25},
  pages        = {714--735},
  year         = {2019}
}

@article{menni2024,
  author       = {Mat{\'\i}as Menni},
  title        = {The successive dimension, without elegance},
  journal      = {Proceedings of the American Mathematical Society},
  volume       = {152},
  number       = {3},
  pages        = {1337--1354},
  year         = {2024},
  doi          = {10.1090/proc/16638}
}

@article{watanabe-junges-rot-hasuo2025,
  author       = {Kazuki Watanabe and Sebastian Junges and Jurriaan Rot and Ichiro Hasuo},
  title        = {A Unifying Approach to Product Constructions for Quantitative Temporal Inference},
  journal      = {Proceedings of the ACM on Programming Languages},
  volume       = {9},
  number       = {OOPSLA1},
  pages        = {1575--1603},
  year         = {2025},
  doi          = {10.1145/3720501}
}
\end{filecontents*}

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\title{AI-generated: Levels and Natural Transformations of the Functor \texorpdfstring{$F_A$}{FA}}

% \author{Ryuya Hora}
% \address{ZEN University, Tokyo, Japan}
% \email{ryuya\_hora@zen.ac.jp}

% \date{\today}
% \subjclass[2020]{18A40, 18B25}
% \keywords{commutative monoid, finitary functor, coskeleton, natural transformation}

\begin{document}

\begin{abstract}
Let $A$ be a commutative monoid, written additively. For a finite set $S$, define
\[
F_A(S)=A^S
\]
and let a map $u:S\to T$ act by summation along the fibers:
\[
(F_A(u)(f))(t)=\sum_{u(s)=t} f(s).
\]
This is the restriction to $\FinSet$ of the usual functor on $\Set$ of finitely supported $A$-valued functions. In this note we compute the level of $F_A$ with respect to the cardinality filtration of $\FinSet$, and we classify all natural transformations $F_A\Rightarrow F_B$. The central point is that $F_A$ is $3$-coskeletal. As a consequence, natural transformations into $F_B$ are forced by arities $1$, $2$, and $3$, and are classified by functions
\[
p:A\times A\to B
\]
satisfying
\[
p(0_A,c)=0_B,
\qquad
p(a+b,c)=p(a,b+c)+p(b,a+c).
\]
We also explain how this description recovers the concrete examples in Kori--Watanabe and yields new examples outside the singly generated case.
\end{abstract}
\maketitle
\memo{AI-generated! Don't believe!!}

\tableofcontents

\section{Introduction}

Let $A$ be a commutative monoid. On the category of sets, one has the finitary functor
\[
\widetilde{F}_A(X)=\{f:X\to A\mid \supp(f)\text{ is finite}\},
\qquad
\supp(f)=\{x\in X\mid f(x)\neq 0_A\}.
\]
Since $\widetilde{F}_A$ is finitary, it is recovered from its restriction to $\FinSet$; see for example \cite{adamek-milius-moss-urbat2015}. Accordingly, throughout the paper we work with the restricted functor
\[
F_A:\FinSet\to\Set.
\]

There are two themes in this note. The first is the cardinality filtration
\[
\FinSet_{\le n}\hookrightarrow \FinSet,
\]
and the associated notions of $n$-skeletal and $n$-coskeletal objects in $\Func{\FinSet}{\Set}$. The second is the explicit description of natural transformations
\[
F_A\Rightarrow F_B.
\]
The link between the two is that $F_B$ turns out to be $3$-coskeletal, so every natural transformation into $F_B$ is controlled by what happens on sets of size at most three.

This viewpoint is particularly convenient when compared with the recent preprint of Kori--Watanabe \cite{kori-watanabe2025}. Their Definition~5 introduces exactly the same family of functors $F_A$, Example~6 identifies the multiset and finite powerset functors inside this family, Proposition~2 shows that any natural transformation $F_A\Rightarrow F_B$ is determined by its component at the $2$-point set, and Theorem~1 together with Examples~7 and~8 computes several singly generated cases explicitly. We shall recover those examples from a single formula for $p(a,c)$ and then use the same formula to produce examples beyond the singly generated setting.

\section{\texorpdfstring{The functor $F_A$}{The functor FA}}

\subsection{Definition and the filtration}

\begin{definition}
Let $A$ be a commutative monoid.
Define a functor
\[
F_A:\FinSet\to\Set
\]
by
\[
F_A(S)=A^S
\]
on objects, and for a map $u:S\to T$ define
\[
(F_A(u)(f))(t)=\sum_{u(s)=t}f(s)
\]
for $f\in A^S$ and $t\in T$.
\end{definition}

\begin{notation}
For $n\ge 0$, let
\[
j_n:\FinSet_{\le n}\hookrightarrow \FinSet
\]
be the full inclusion.
For a functor $H:\FinSet\to\Set$, define
\[
\sk_n H:=\Lan_{j_n}(j_n^*H),
\qquad
\cosk_n H:=\Ran_{j_n}(j_n^*H).
\]
Pointwise, one has
\[
(\sk_n H)(X)\cong \colim_{(u:S\to X)\in (j_n\downarrow X)} H(S),
\]
and
\[
(\cosk_n H)(X)\cong \lim_{(u:X\to S)\in (X\downarrow j_n)} H(S).
\]
\end{notation}

\begin{remark}
If $X$ is a finite set, an element of $(\cosk_3 F_A)(X)$ is concretely a compatible family
\[
(\lambda_u)_{u:X\to S,\ \abs{S}\le 3},
\qquad
\lambda_u\in A^S,
\]
compatible under postcomposition. Intuitively, this means that for every way of cutting $X$ into at most three pieces, we are given the sums on those pieces, and these sums agree whenever one partition is obtained from another by merging pieces.
\end{remark}

\subsection{\texorpdfstring{The level of $F_A$}{The level of FA}}

We now spell out the previous remark in concrete pictures.

\begin{example}\label{ex:partition-2}
Let $X=\{1,2,3,4\}$ and let $T=\{1,4\}$.
The map
\[
\chi_T:X\to 2
\]
remembers only the decomposition of $X$ into the two parts $T$ and $X\setminus T$. If $f\in F_A(X)=A^X$ corresponds to values $a_1,a_2,a_3,a_4\in A$, then
\[
F_A(\chi_T)(f)=(a_1+a_4,\ a_2+a_3).
\]
Thus a $2$-partition only records subset sums.
\end{example}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm, every node/.style={font=\small}]
\node[draw, rounded corners, fill=blue!7, minimum width=3.6cm, minimum height=1cm] (T) at (0,0) {$T$};
\node[draw, rounded corners, fill=green!7, minimum width=3.6cm, minimum height=1cm] (C) at (4.8,0) {$X\setminus T$};
\foreach \x in {-1.2,-0.5,0.2,0.9} {\fill (\x,0) circle (1.7pt);}
\foreach \x in {3.6,4.3,5.0,5.7} {\fill (\x,0) circle (1.7pt);}
\draw[->,thick] (0,-0.7) -- (0,-2.0);
\draw[->,thick] (4.8,-0.7) -- (4.8,-2.0);
\node at (0,-2.35) {$1$};
\node at (4.8,-2.35) {$2$};
\node at (2.4,-2.35) {$\chi_T:X\to 2$};
\end{tikzpicture}
\caption{A map $X\to 2$ remembers a subset and its complement.}
\end{figure}

\begin{example}\label{ex:partition-3}
Fix $x\in T\subseteq X$.
The map
\[
u:X\to 3
\]
with fibers
\[
\{x\},\qquad T\setminus\{x\},\qquad X\setminus T
\]
refines the previous $2$-partition by separating one point from the rest of $T$.
If $f\in A^X$ corresponds to $(a_y)_{y\in X}$, then
\[
F_A(u)(f)=\left(a_x,\ \sum_{y\in T\setminus\{x\}} a_y,\ \sum_{y\in X\setminus T} a_y\right).
\]
This is exactly the configuration used in the proof of $3$-coskeletality.
\end{example}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm, every node/.style={font=\small}]
\node[draw, rounded corners, fill=red!8, minimum width=1.4cm, minimum height=1cm] (x) at (0,0) {$\{x\}$};
\node[draw, rounded corners, fill=blue!8, minimum width=3.2cm, minimum height=1cm] (tx) at (3.3,0) {$T\setminus\{x\}$};
\node[draw, rounded corners, fill=green!8, minimum width=3.6cm, minimum height=1cm] (c) at (7.8,0) {$X\setminus T$};
\fill (-0.25,0) circle (1.7pt);
\foreach \x in {2.4,3.1,3.8,4.5} {\fill (\x,0) circle (1.7pt);}
\foreach \x in {6.6,7.3,8.0,8.7} {\fill (\x,0) circle (1.7pt);}
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\draw[->,thick] (3.3,-0.7) -- (3.3,-2.0);
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\node at (0,-2.35) {$1$};
\node at (3.3,-2.35) {$2$};
\node at (7.8,-2.35) {$3$};
\node at (4.0,-2.35) {$u:X\to 3$};
\end{tikzpicture}
\caption{A $3$-partition isolates one point and forces additivity on subsets.}
\end{figure}

\begin{figure}[ht]
\centering
\[
\begin{tikzcd}[column sep=huge,row sep=large]
& X \arrow[dl, "\chi_{\{x\}}"'] \arrow[d, "u"] \arrow[dr, "\chi_T"] & \\
2 & 3 \arrow[l, "r_1"] \arrow[r, "m"'] & 2
\end{tikzcd}
\]
\caption{The maps used to read off $\mu(\{x\})$ and $\mu(T)$ from the same $3$-partition.}
\end{figure}

\begin{proposition}\label{prop:3cosk}
For every commutative monoid $A$, the functor $F_A$ is $3$-coskeletal.
\end{proposition}

\begin{proof}
Let $X$ be a finite set.
If $\abs{X}\le 3$, then the identity $X\to X$ is an initial object of $(X\downarrow j_3)$, so the canonical map
\[
F_A(X)\to (\cosk_3 F_A)(X)
\]
is automatically a bijection.
Thus we may assume $\abs{X}>3$.

Let $(\lambda_u)$ be an element of $(\cosk_3 F_A)(X)$.
Thus for each map
\[
u:X\to S,
\qquad \abs{S}\le 3,
\]
we are given an element $\lambda_u\in A^S$, and these are compatible under postcomposition.

For each subset $T\subseteq X$, let
\[
\chi_T:X\to 2
\]
be the characteristic map with $\chi_T^{-1}(1)=T$.
Write
\[
\mu(T)=\text{the first coordinate of }\lambda_{\chi_T}\in A^2.
\]
For each $x\in X$, define
\[
a_x:=\mu(\{x\}).
\]

We claim that for every subset $T\subseteq X$,
\[
\mu(T)=\sum_{x\in T} a_x.
\]
We prove this by induction on $\abs{T}$.

If $T=\varnothing$, then $\chi_\varnothing$ factors as
\[
X\to 1 \xrightarrow{i_2} 2,
\]
where $i_2$ lands in the second point. Hence
\[
\lambda_{\chi_\varnothing}=F_A(i_2)(\lambda_{X\to 1}),
\]
so its first coordinate is $0_A$. Therefore $\mu(\varnothing)=0_A$.

If $\abs{T}=1$, this is the definition of $a_x$.

Assume now that $\abs{T}\ge 2$, and choose $x\in T$.
Let
\[
u:X\to 3
\]
be the map whose fibers are
\[
\{x\},\qquad T\setminus\{x\},\qquad X\setminus T.
\]
Write
\[
\lambda_u=(b_1,b_2,b_3)\in A^3.
\]
Let $r_1,r_2,m:3\to 2$ be defined by
\[
r_1^{-1}(1)=\{1\},
\qquad
r_2^{-1}(1)=\{2\},
\qquad
m^{-1}(1)=\{1,2\}.
\]
Then
\[
r_1\circ u=\chi_{\{x\}},
\qquad
r_2\circ u=\chi_{T\setminus\{x\}},
\qquad
m\circ u=\chi_T.
\]
By compatibility we obtain
\[
F_A(r_1)(b_1,b_2,b_3)=\lambda_{\chi_{\{x\}}},
\qquad
F_A(r_2)(b_1,b_2,b_3)=\lambda_{\chi_{T\setminus\{x\}}},
\qquad
F_A(m)(b_1,b_2,b_3)=\lambda_{\chi_T}.
\]
Taking first coordinates gives
\[
b_1=\mu(\{x\})=a_x,
\qquad
b_2=\mu(T\setminus\{x\}),
\qquad
b_1+b_2=\mu(T).
\]
Therefore
\[
\mu(T)=a_x+\mu(T\setminus\{x\}).
\]
By the induction hypothesis,
\[
\mu(T)=a_x+\sum_{y\in T\setminus\{x\}}a_y=
\sum_{y\in T}a_y.
\]
This proves the claim.

Now let $f\in A^X$ be the function defined by
\[
f(x)=a_x.
\]
We show that the compatible family $(\lambda_u)$ is exactly the image of $f$ under the canonical map
\[
F_A(X)\to (\cosk_3F_A)(X).
\]

If $u:X\to 3$ has fibers
\[
T_1=u^{-1}(1),
\qquad
T_2=u^{-1}(2),
\qquad
T_3=u^{-1}(3),
\]
and if
\[
\lambda_u=(c_1,c_2,c_3),
\]
then for each $i=1,2,3$ let $r_i:3\to 2$ be the map sending $i$ to $1$ and the other two points to $2$.
Since $r_i\circ u=\chi_{T_i}$, compatibility gives
\[
F_A(r_i)(c_1,c_2,c_3)=\lambda_{\chi_{T_i}}.
\]
Taking first coordinates yields
\[
c_i=\mu(T_i)=\sum_{x\in T_i} a_x.
\]
Hence
\[
\lambda_u=
\left(
\sum_{x\in T_1}a_x,
\sum_{x\in T_2}a_x,
\sum_{x\in T_3}a_x
\right)
=F_A(u)(f).
\]
The same argument works for maps $X\to 2$ and for the unique map $X\to 1$.
Therefore $(\lambda_u)$ is induced by $f$.

Uniqueness is immediate because the characteristic maps of singletons recover each $a_x$.
Thus the canonical map
\[
F_A(X)\to (\cosk_3F_A)(X)
\]
is a bijection for every finite set $X$.
\end{proof}

\begin{corollary}\label{cor:restriction}
Let $B$ be a commutative monoid and let $H:\FinSet\to\Set$ be any functor.
Then restriction along $j_3$ induces a bijection
\[
\Nat(H,F_B)\cong \Nat(j_3^*H,j_3^*F_B).
\]
In particular, every natural transformation into $F_B$ is determined by its components on sets of size at most $3$.
\end{corollary}

\begin{proof}
By \cref{prop:3cosk}, the functor $F_B$ is $3$-coskeletal, so
\[
F_B\cong \cosk_3(j_3^*F_B).
\]
Since $j_3^*$ has right adjoint $\cosk_3$, we obtain
\[
\Nat(H,F_B)
\cong
\Nat(H,\cosk_3(j_3^*F_B))
\cong
\Nat(j_3^*H,j_3^*F_B).
\qedhere
\]
\end{proof}

\begin{remark}\label{rem:lower-levels}
If $A$ is nontrivial, then $F_A$ is not $n$-skeletal for any finite $n$, and it is not $2$-coskeletal. These two statements are useful for orientation, but they are not needed for the classification of natural transformations. We therefore move their proofs to \Cref{sec:appendix-lower}.
\end{remark}

\section{Natural transformations}

\subsection{An additive identity}

\begin{lemma}\label{lem:main-identity}
Let $A$ and $B$ be commutative monoids, and let
\[
p:A\times A\to B
\]
be a function satisfying
\[
p(0_A,c)=0_B,
\qquad
p(a+b,c)=p(a,b+c)+p(b,a+c)
\]
for all $a,b,c\in A$.
Then for every finite family $a_1,\dots,a_n\in A$ and every $c\in A$,
\[
p(a_1+\cdots+a_n,c)
=
\sum_{i=1}^n p\!\left(a_i,\ c+\sum_{j\neq i}a_j\right).
\]
\end{lemma}

\begin{proof}
We argue by induction on $n$.
If $n=0$, the statement is $p(0_A,c)=0_B$.
Assume the statement holds for $n$.
Then
\[
p(a_1+\cdots+a_n+a_{n+1},c)
=
p(a_1+\cdots+a_n,c+a_{n+1})+p(a_{n+1},c+a_1+\cdots+a_n).
\]
Applying the induction hypothesis to the first term gives the desired formula for $n+1$.
\end{proof}

\subsection{Classification theorem}

\begin{theorem}\label{thm:classification}
Let $A$ and $B$ be commutative monoids.
Then natural transformations
\[
\eta:F_A\Rightarrow F_B
\]
are in bijection with functions
\[
p:A\times A\to B
\]
satisfying
\[
p(0_A,c)=0_B,
\qquad
p(a+b,c)=p(a,b+c)+p(b,a+c)
\]
for all $a,b,c\in A$.

The correspondence is given as follows.
\begin{enumerate}
\item Given a natural transformation $\eta$, define
\[
p_\eta(a,b)=\bigl(\eta_2(a,b)\bigr)(1).
\]
\item Given a function $p$ satisfying the two identities above, define
\[
\eta^p_X:A^X\to B^X
\]
by
\[
(\eta^p_X(f))(x)=p\!\left(f(x),\sum_{y\neq x}f(y)\right).
\]
\end{enumerate}
These two constructions are inverse to each other.
\end{theorem}

\begin{proof}
Let $\eta:F_A\Rightarrow F_B$ be a natural transformation, and define
\[
p(a,b):=p_\eta(a,b)=\bigl(\eta_2(a,b)\bigr)(1).
\]

\para{Step 1: the component on $2$}
Let $\tau:2\to 2$ be the transposition.
Naturality with respect to $\tau$ gives
\[
\eta_2(a,b)=\bigl(p(a,b),p(b,a)\bigr).
\]
Now let $i_2:1\to 2$ be the injection landing in the second point.
Since
\[
F_A(i_2)(c)=(0_A,c),
\qquad
F_B(i_2)(d)=(0_B,d),
\]
naturality gives
\[
\eta_2(0_A,c)=F_B(i_2)(\eta_1(c))=(0_B,\eta_1(c)).
\]
Comparing the first coordinates, we obtain
\[
p(0_A,c)=0_B.
\]
Comparing the second coordinates, we obtain
\[
\eta_1(c)=p(c,0_A).
\]

\para{Step 2: the component on $3$}
Write
\[
\eta_3(a,b,c)=(u,v,w)\in B^3.
\]
Let $r_1,r_2,r_3:3\to 2$ be the maps isolating the first, second, and third points:
\[
r_1^{-1}(1)=\{1\},
\qquad
r_2^{-1}(1)=\{2\},
\qquad
r_3^{-1}(1)=\{3\}.
\]
Then
\[
F_A(r_1)(a,b,c)=(a,b+c),
\qquad
F_A(r_2)(a,b,c)=(b,a+c),
\qquad
F_A(r_3)(a,b,c)=(c,a+b).
\]
Naturality implies
\[
F_B(r_1)(u,v,w)=\eta_2(a,b+c),
\qquad
F_B(r_2)(u,v,w)=\eta_2(b,a+c),
\qquad
F_B(r_3)(u,v,w)=\eta_2(c,a+b).
\]
Taking first coordinates gives
\[
u=p(a,b+c),
\qquad
v=p(b,a+c),
\qquad
w=p(c,a+b).
\]
Therefore
\[
\eta_3(a,b,c)=\bigl(p(a,b+c),p(b,a+c),p(c,a+b)\bigr).
\]

\para{Step 3: the relation on $p$}
Let $m:3\to 2$ be the map with fibers
\[
m^{-1}(1)=\{1,2\},
\qquad
m^{-1}(2)=\{3\}.
\]
Then
\[
F_A(m)(a,b,c)=(a+b,c),
\qquad
F_B(m)(u,v,w)=(u+v,w).
\]
Naturality gives
\[
\eta_2(a+b,c)=F_B(m)(\eta_3(a,b,c)).
\]
Comparing the first coordinates yields
\[
p(a+b,c)=p(a,b+c)+p(b,a+c).
\]
Thus every natural transformation yields a function $p$ satisfying the required identities.

\para{Step 4: construction from $p$}
Conversely, suppose that
\[
p:A\times A\to B
\]
satisfies
\[
p(0_A,c)=0_B,
\qquad
p(a+b,c)=p(a,b+c)+p(b,a+c).
\]
Define
\[
\eta^p_X(f)(x):=p\!\left(f(x),\sum_{y\neq x}f(y)\right).
\]
We prove that $\eta^p$ is natural.
Let $u:X\to Y$ be a map, let $f\in A^X$, and let $y\in Y$.
Write
\[
S=u^{-1}(y),
\qquad
c=\sum_{z\notin S} f(z).
\]
Then
\[
(F_B(u)(\eta^p_X(f)))(y)
=
\sum_{x\in S} p\!\left(f(x),\ c+\sum_{\substack{x'\in S\\ x'\neq x}} f(x')\right).
\]
By \cref{lem:main-identity}, this is equal to
\[
p\!\left(\sum_{x\in S}f(x),c\right).
\]
On the other hand,
\[
(F_A(u)(f))(y)=\sum_{x\in S}f(x),
\qquad
\sum_{y'\neq y}(F_A(u)(f))(y')=\sum_{z\notin S}f(z)=c.
\]
Hence
\[
(F_B(u)(\eta^p_X(f)))(y)=(\eta^p_Y(F_A(u)(f)))(y).
\]
So $\eta^p$ is natural.

\para{Step 5: the two constructions are inverse}
If we start with $p$, then
\[
\eta^p_2(a,b)=\bigl(p(a,b),p(b,a)\bigr),
\]
so $p_{\eta^p}(a,b)=p(a,b)$.
Conversely, if we start with $\eta$, then for any finite $X$, any $f\in A^X$, and any $x\in X$, the characteristic map
\[
\chi_{\{x\}}:X\to 2
\]
gives
\[
F_A(\chi_{\{x\}})(f)=\left(f(x),\sum_{y\neq x}f(y)\right).
\]
Naturality yields
\[
F_B(\chi_{\{x\}})(\eta_X(f))=
\eta_2\!\left(f(x),\sum_{y\neq x}f(y)\right).
\]
Taking the first coordinate, we obtain
\[
(\eta_X(f))(x)=p_\eta\!\left(f(x),\sum_{y\neq x}f(y)\right)=
(\eta^{p_\eta}_X(f))(x).
\]
Hence $\eta=\eta^{p_\eta}$.
\end{proof}

\begin{remark}
The theorem shows that the $2$-point set contributes the binary datum $p(a,c)$, while the $3$-point set contributes exactly the cocycle identity
\[
p(a+b,c)=p(a,b+c)+p(b,a+c).
\]
By \cref{cor:restriction}, nothing new appears in higher arity.
\end{remark}

\subsection{Examples and comparison with Kori--Watanabe}

We now explain how the concrete examples in Kori--Watanabe are recovered from \cref{thm:classification}. The relevant references are their Definition~5, Example~6, Proposition~2, Theorem~1, Examples~7--8, Lemma~3, and Proposition~3 \cite{kori-watanabe2025}.

\begin{example}[Multisets to powersets and additive weights]\label{ex:KW-M-to-Pf}
Let $M=F_{\N}$, and let $B$ be any commutative monoid. Given a natural transformation
\[
\eta:F_{\N}\Rightarrow F_B,
\]
let $p:\N\times \N\to B$ be its associated function.
Define
\[
b(0)=0_B,
\qquad
b(s)=p(1,s-1)
\quad (s\ge 1).
\]
Then an induction on $n$ using \cref{thm:classification} shows that
\[
p(n,m)=n\cdot b(n+m)
\]
for all $n,m\in\N$.
Therefore
\[
(\eta_X(f))(x)=f(x)\cdot b\!\left(\sum_{y\in X} f(y)\right).
\]

\begin{enumerate}
\item If $B=\B$ with idempotent addition, then $n\cdot b=b$ for $n>0$, so
\[
(\eta_X(f))(x)=
\begin{cases}
b\!\left(\sum_{y\in X} f(y)\right),& f(x)>0,\\
0,& f(x)=0.
\end{cases}
\]
Equivalently,
\[
\eta_X(f)=\{x\in X\mid f(x)>0\ \text{and}\ b(\sum_{y\in X}f(y))=1\}.
\]
This is exactly the shape of Kori--Watanabe, Example~7(1).

\item If $B=\R_{\ge 0}$ under addition, then
\[
(\eta_X(f))(x)=f(x)\,b\!\left(\sum_{y\in X}f(y)\right),
\]
which is exactly their Example~7(2).
\end{enumerate}
\end{example}

\begin{example}[Finite powersets to additive targets]\label{ex:KW-Pf-to-additive}
Let $P_f=F_{\B}$, where $\B=\{0,1\}$ is viewed as the idempotent commutative monoid with $1+1=1$.
Let
\[
\eta:F_{\B}\Rightarrow F_B
\]
be a natural transformation, and let $q(c)=p(1,c)$.
Since $1+1=1$ in the source monoid, the identity in \cref{thm:classification} gives
\[
q(c)=p(1,c)=p(1,1+c)+p(1,1+c)=2\cdot q(1+c).
\]

\begin{enumerate}
\item If $B=\N$ or $B=\R_{\ge 0}$, then the only solution is $q(c)=0$ for all $c$.
Hence the only natural transformation
\[
P_f\Rightarrow M
\qquad\text{or}\qquad
P_f\Rightarrow F_{\R_{\ge 0}}
\]
is the zero transformation. This is Kori--Watanabe, Example~8(1).

\item Kori--Watanabe also compute the case
\[
P_f\Rightarrow F_{(\R_{\ge 0},\cdot,1)}
\]
in their Example~8(2). Since that target is naturally multiplicative rather than additive, it sits slightly outside the notation of the present note; nevertheless it is the same calculation after rewriting the target monoid multiplicatively.
\end{enumerate}
\end{example}

\begin{example}[A quick derivation of Kori--Watanabe, Proposition~3(4)]\label{ex:KW-subdistribution}
Kori--Watanabe, Proposition~3(4), describes natural transformations
\[
M\Rightarrow D_{\le 1},
\]
where $D_{\le 1}$ is the finite subdistribution functor.
Since $D_{\le 1}$ is a subfunctor of $F_{\R_{\ge 0}}$, \cref{ex:KW-M-to-Pf} says that any natural transformation into $F_{\R_{\ge 0}}$ has the form
\[
(\eta_X(f))(x)=f(x)\,b\!\left(\sum_{y\in X} f(y)\right).
\]
To land in $D_{\le 1}(X)$, we must have
\[
\sum_{x\in X}(\eta_X(f))(x)
=
\left(\sum_{x\in X}f(x)\right)b\!\left(\sum_{x\in X}f(x)\right)
\le 1.
\]
Thus if $s>0$ and we set
\[
c_s:=s\,b(s)\in [0,1],
\]
then
\[
(\eta_X(f))(x)=
\begin{cases}
\dfrac{f(x)}{\sum_{y\in X}f(y)}\,c_{\sum_{y\in X}f(y)},& \sum_{y\in X}f(y)>0,\\
0,& \sum_{y\in X}f(y)=0.
\end{cases}
\]
This is exactly the formula stated in Proposition~3(4), equivalently Corollary~1, of \cite{kori-watanabe2025}.
\end{example}

\begin{example}[A new example beyond the singly generated case]\label{ex:new-N2-to-N}
Let
\[
A=\N^2,
\qquad
B=\N.
\]
Fix any function
\[
b:\N^2\to\N
\]
with $b(0,0)=0$, and define
\[
p\bigl((a_1,a_2),(c_1,c_2)\bigr)
:=
(a_1+2a_2)\,b(a_1+c_1,a_2+c_2).
\]
Then $p(0,c)=0$, and for $a,a',c\in \N^2$ one has
\[
p(a+a',c)=p(a,a'+c)+p(a',a+c)
\]
because $a\mapsto a_1+2a_2$ is additive.
Hence \cref{thm:classification} yields a natural transformation
\[
F_{\N^2}\Rightarrow F_{\N}
\]
given by
\[
(\eta_X(f))(x)
=
\bigl(f_1(x)+2f_2(x)\bigr)
\,b\!\left(\sum_{y\in X}f_1(y),\sum_{y\in X}f_2(y)\right).
\]
This example is genuinely outside the singly generated framework of Kori--Watanabe, Theorem~1, because the source monoid $\N^2$ is not singly generated.
\end{example}

\begin{example}[A new support-threshold example]\label{ex:new-N2-to-bool}
Let $A=\N^2$ and let the target be the idempotent monoid $\B$.
Fix a subset $R\subseteq \N^2$ with $(0,0)\notin R$, and define
\[
p\bigl((a_1,a_2),(c_1,c_2)\bigr)=
\begin{cases}
1,& (a_1,a_2)\neq (0,0)\ \text{and}\ (a_1+c_1,a_2+c_2)\in R,\\
0,& \text{otherwise}.
\end{cases}
\]
Then \cref{thm:classification} gives a natural transformation
\[
F_{\N^2}\Rightarrow P_f
\]
whose value on $f=(f_1,f_2):X\to \N^2$ is
\[
\eta_X(f)=
\left\{x\in X\ \middle|\ f(x)\neq (0,0)
\ \text{and}\ \left(\sum_{y\in X}f_1(y),\sum_{y\in X}f_2(y)\right)\in R\right\}.
\]
Thus our formula packages a whole family of ``select the support once the total coloured mass enters a prescribed region'' operations.
\end{example}

\appendix

\section{A brief bibliographic survey}

This appendix records only references that are especially close to the two themes of the note: the functor $F_A$ itself, and levels/coskeleta on categories built from finite sets.

\subsection{\texorpdfstring{The functor $F_A$ and its natural transformations}{The functor FA and its natural transformations}}

The functor of finitely supported commutative-monoid-valued functions appears naturally in coalgebraic treatments of weighted or monoid-labeled transition systems; a classical reference is Gumm--Schr"oder \cite{gumm-schroeder2001}. The point of view that a finitary endofunctor on $\Set$ is controlled by its restriction to $\FinSet$ is standard; see Ad\'amek--Milius--Moss--Urbat \cite{adamek-milius-moss-urbat2015}. This is the background for treating $F_A$ as an object of $\Func{\FinSet}{\Set}$.

On the side of natural transformations between branching functors, Dahlqvist--Neves \cite{dahlqvist-neves2018} classify many operations of the form
\[
T^n\Rightarrow T
\]
for powerset- and multiset-type monads. The closest recent reference to the present note is Kori--Watanabe \cite{kori-watanabe2025}. In the current arXiv preprint, the relevant points for us are Definition~5, Example~6, Proposition~2, Theorem~1, Examples~7--8, Lemma~3, and Proposition~3.

\subsection{Levels, skeleta, and coskeleta}

The general language of levels and essential localizations goes back to Kelly--Lawvere \cite{kelly-lawvere1989}. For explicit calculations in familiar presheaf toposes, a standard reference is Kennett--Riehl--Roy--Zaks \cite{kennett-riehl-roy-zaks2011}. For more structural developments of skeleta and dimension-like operations in toposes, see Menni \cite{menni2019,menni2024}. A recent nearby example in a finite-set-based topos is the work of Hora--Kamio--Maehara \cite{hora-kamio-maehara2025}.

\section{Auxiliary proofs on lower levels}\label{sec:appendix-lower}

\begin{proposition}
If $A$ is nontrivial, then $F_A$ is not $n$-skeletal for any finite $n$.
\end{proposition}

\begin{proof}
Fix $n\ge 0$, and choose $a\in A$ with $a\neq 0_A$.
Let $X=n+1$, and consider the constant function
\[
f:X\to A,
\qquad
f(x)=a.
\]
Every element of $(\sk_nF_A)(X)$ is represented by some pair $(u,g)$ with $u:S\to X$, $\abs{S}\le n$, and $g\in A^S$.
Its image in $F_A(X)=A^X$ is $F_A(u)(g)$, and this function vanishes outside $u(S)$.
Hence it can be nonzero at at most $n$ points of $X$.
But $f$ is nonzero at all $n+1$ points.
Therefore $f$ does not lie in the image of
\[
(\sk_nF_A)(X)\to F_A(X),
\]
so $F_A$ is not $n$-skeletal.
\end{proof}

\begin{proposition}
If $A$ is nontrivial, then $F_A$ is not $2$-coskeletal.
\end{proposition}

\begin{proof}
Choose $a\in A$ with $a\neq 0_A$, and let $X=\{1,2,3\}$.
Define
\[
\mu:\Pow(X)\to A
\]
by
\[
\mu(T)=
\begin{cases}
0_A,& \abs{T}=0 \text{ or }1,\\
a,& \abs{T}=2 \text{ or }3.
\end{cases}
\]
For each map $u:X\to S$ with $\abs{S}\le 2$, define $\lambda_u\in A^S$ as follows.
If $S=1$, put $\lambda_u=(a)$.
If $S=2$, let $T=u^{-1}(1)$ and put
\[
\lambda_u=(\mu(T),\mu(X\setminus T)).
\]
Exactly as in the proof of \cref{prop:3cosk}, one checks that $(\lambda_u)$ is compatible, hence defines an element of $(\cosk_2F_A)(X)$.

Suppose that it comes from some $f=(f_1,f_2,f_3)\in A^3$.
For each $i\in X$, let $\chi_{\{i\}}:X\to 2$ be the characteristic map of the singleton $\{i\}$.
Then
\[
F_A(\chi_{\{i\}})(f)=\lambda_{\chi_{\{i\}}}=(0_A,a).
\]
Therefore $f_i=0_A$ for all $i$.
But then the image of $f$ under the unique map $X\to 1$ is $(0_A)$, whereas by construction $\lambda_{X\to 1}=(a)$.
This is a contradiction.
Therefore $F_A$ is not $2$-coskeletal.
\end{proof}


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\end{document}