\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*} \documentclass{amsart} \usepackage[left=2cm, right=2cm]{geometry} \usepackage[utf8]{inputenc} \usepackage{amsfonts, amsthm, amssymb, mathtools,etoolbox} \usepackage{blindtext} \usepackage[colorlinks=true, urlcolor=blue, linkcolor=blue, citecolor=blue]{hyperref} \usepackage{tikz,tikz-cd} \usepackage{cleveref} \usepackage{xcolor} \usepackage{array} \usepackage[style=alphabetic,sorting=nyt, maxnames=4]{biblatex} % \usepackage[style=authoryear, maxnames=4]{biblatex} \renewbibmacro{in:}{} \addbibresource{fa_refs.bib} \tikzset{pullback/.style={minimum size=1.2ex,path picture={ \draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);% }}} \usetikzlibrary{calc,arrows.meta,positioning} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[subsection] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{todo}[theorem]{Todo} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{fact}[theorem]{Fact} \newtheorem{claim}{Claim}[theorem] \crefname{claim}{Claim}{Claims} \renewcommand{\theclaim}{\thetheorem.\alph{claim}} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{notation}[theorem]{Notation} \newtheorem{puzzle}[theorem]{Puzzle} \newtheorem{idea}[theorem]{Idea} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\dq}[1]{``#1"} \newcommand{\memo}[1]{\textcolor{red}{memo: #1}} % \newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}} \newcommand{\invmemo}[1]{} \newcommand{\horamemo}[1]{\textcolor{green!70!black}{hora: #1}} \newcommand{\para}[1]{\par\medskip\noindent\textbf{#1}\enspace} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\B}{\mathbb{B}} \newcommand{\C}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathbf{F}} \renewcommand{\L}{\mathcal{L}} \newcommand{\id}{\mathrm{id}} \newcommand{\op}{\mathrm{op}} \newcommand{\ob}{\mathrm{ob}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\sSet}{\mathbf{sSet}} \newcommand{\FinSet}{\mathbf{FinSet}} \newcommand{\PSh}{\mathbf{PSh}} \newcommand{\Sh}{\mathbf{Sh}} \newcommand{\Cont}{\mathbf{Cont}} \newcommand{\Func}[2]{[#1,#2]} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\demph}[1]{\textit{#1}} \font\maljapanese=dmjhira at 2.5ex \newcommand{\yo}{\textrm{\!\maljapanese\char"48}} \newcommand{\Pow}{\mathcal{P}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Nat}{\operatorname{Nat}} \newcommand{\Lan}{\operatorname{Lan}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\sk}{\operatorname{sk}} \newcommand{\cosk}{\operatorname{cosk}} \DeclareMathOperator*{\colim}{colim} \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);} \draw[->,thick] (0,-0.7) -- (0,-2.0); \draw[->,thick] (3.3,-0.7) -- (3.3,-2.0); \draw[->,thick] (7.8,-0.7) -- (7.8,-2.0); \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} \printbibliography \end{document}