1. Thesis and contribution
Thesis (T). Operators of technology platforms ("technocrats") hold de facto political influence. It is determined by their platforms and by how many people pay for their services. It is a purely utilitarian function of the number of network participants, and it acts on political opinion.
We give (T) a precise semantics and isolate what it presupposes. We do not test (T) empirically; we classify the structure it asserts.
1. A *polity* is a span of $\pi$-finite spaces from persons to platforms (§2). Decategorification at truncation levels $-1$, $0$ and $\infty$ counts persons, memberships and payments. These are functors to matrices over $\mathbb B$, $\mathbb N$ and $\mathbb Q_{\ge0}$. 2. "Each to count for one" is the universal property of $\mathrm{Fin}^{\simeq}$ as the free symmetric monoidal $\infty$-groupoid on a point (Theorem 3.1). Utilitarian influence is forced to be a multiple of the count. Payment counts are characterized by additivity and multiplicativity (Proposition 3.4). 3. In a linear attention model the operator's opinion share is $N_p/|U|$ under single-homing and absence of selection (Proposition 4.1). Counterexamples show neither hypothesis can be dropped. 4. Bloc influence is the cardinality of the realization of a Čech nerve. It is given by inclusion–exclusion and is subadditive (Theorem 5.1). 5. Person counts do not compose, because $\mathbb B\to\mathbb N$ is not a semiring map. Hence reach through social relations is not a function of participant counts (Proposition 6.1).
Conclusion: (T) holds exactly on the part of the political structure freely generated by anonymous, non-overlapping, non-transmitting participants. Elsewhere the count is a decategorified shadow of the polity.
Conventions. $\mathcal S$ is the $\infty$-category of spaces. A space is $\pi$-finite if it has finitely many components and finitely many nontrivial homotopy groups, all finite. Pullbacks are homotopy pullbacks. The homotopy cardinality [1,3] of a $\pi$-finite $X$ is $$ |X|=\sum_{[x]\in\pi_0X}\ \prod_{n\ge1}|\pi_n(X,x)|^{(-1)^n}\in\mathbb Q_{\ge0}. $$ It is equivalence invariant, $|X\sqcup Y|=|X|+|Y|$, $|X\times Y|=|X|\,|Y|$, $|\varnothing|=0$, and $|X|>0$ if $X\neq\varnothing$. $\tau_{\le k}$ is $k$-truncation; $\tau_{\le-1}X$ is $\varnothing$ or $*$ (propositional truncation [9]). $\mathbb B=(\{0,1\},\vee,\wedge)$ is the Boolean semiring.
2. Polities and their three counts
Definition 2.1 (Polity). A polity consists of a finite set $U$ of persons, a finite set $\Pi$ of platforms, and a $\pi$-finite space $E$ with a map $E\to U\times\Pi$. The fibre $E_{u,p}$ is the space of paid participations of $u$ in $p$. Points are paid memberships, paths identify memberships (for example merged accounts), and higher cells are coherences between identifications. Put $S_p=\{u:E_{u,p}\neq\varnothing\}$ and $\Pi_u=\{p:E_{u,p}\neq\varnothing\}$.
Definition 2.2 (Counts). For $p\in\Pi$, $$ N_p=\sum_u|\tau_{\le-1}E_{u,p}|=|S_p|,\qquad A_p=\sum_u|\pi_0E_{u,p}|,\qquad R_p=\sum_u|E_{u,p}|. $$ $N_p$ counts persons who pay and $A_p$ counts paid memberships. $R_p$ counts payment units under the following encoding. A unit shared by $k$ persons contributes $B(\mathbb Z/k)$ to each sharer's fibre; its cardinality is $1/k$, so the $k$ sharers contribute one unit together. Two merged accounts form a contractible fibre: one membership, one unit.
Proposition 2.3. (a) $N_p\le A_p$, with equality iff every fibre $E_{u,p}$ has at most one component. (b) $R_p$ is not determined by $(N_p,A_p)$. (c) No cardinality detects contractibility: $X=B(\mathbb Z/2)\times K(\mathbb Z/2,2)$ is connected, not contractible, and $|X|=\tfrac12\cdot 2=1=|*|$.
*Proof.* (a) Termwise $|\tau_{\le-1}Y|\le|\pi_0Y|$, with equality iff $|\pi_0Y|\le1$. (b) Fibres $*$ and $B(\mathbb Z/2)$ have equal $N,A$ but $R=1$ and $R=\tfrac12$. (c) Product formula. $\square$
So (T) is ambiguous until a truncation level is fixed. "How many people pay" is $N_p$; "how many subscriptions" is $A_p$; "how much is paid" is $R_p$.
Definition 2.4 (The category $\mathbf{Pol}$). Objects are finite sets. A morphism $V\to W$ is a $\pi$-finite space $F$ over $V\times W$, up to equivalence over $V\times W$. The composite of $F:V\to W$ and $G:W\to X$ is $F\times_WG$ over $V\times X$; identities are diagonals. This is the homotopy category of spans of $\pi$-finite spaces [2,3], restricted to discrete objects. Since $W$ is discrete, $$ (G\circ F)_{v,x}\simeq\coprod_{w\in W}F_{v,w}\times G_{w,x}. \tag{2.1} $$ A polity is a morphism $E:U\to\Pi$.
Proposition 2.5 (Three decategorifications). The assignments $F\mapsto(|F_{v,w}|)$, $F\mapsto(|\pi_0F_{v,w}|)$ and $F\mapsto(|\tau_{\le-1}F_{v,w}|)$ are functors from $\mathbf{Pol}$ to matrices over $\mathbb Q_{\ge0}$, $\mathbb N$ and $\mathbb B$. The support map $\sigma(x)=[x>0]$ is a semiring homomorphism $\mathbb Q_{\ge0}\to\mathbb B$, and $\sigma|F_{v,w}|=|\tau_{\le-1}F_{v,w}|$.
*Proof.* Each of $|{-}|$, $|\pi_0(-)|$ and $\tau_{\le-1}$ is equivalence invariant and sends finite coproducts to sums (disjunction) and products to products (conjunction). Apply this to (2.1). Diagonal fibres are $*$ or $\varnothing$, giving identity matrices. A sum or product of nonnegative rationals is positive iff some summand, respectively every factor, is positive. Finally $|Y|>0$ iff $Y\neq\varnothing$. $\square$
Let $\mathbf 1_U:*\to U$ be $U$ itself. Then $R_p=|E\circ\mathbf 1_U|_{*,p}$ and $A_p=|\pi_0(E\circ\mathbf 1_U)_{*,p}|$: payments and memberships are fibre counts of a composite. But $|\tau_{\le-1}(E\circ\mathbf 1_U)_{*,p}|=[N_p>0]$. The person count $N_p$ applies the map $\mathbb B\to\mathbb N$, $b\mapsto b$, which is not additive ($1\vee1=1$). This is the source of §6.
3. Each to count for one: a universal property
Let $\mathrm{Fin}^{\simeq}\simeq\coprod_{n\ge0}B\Sigma_n$ be the groupoid of finite sets and bijections, symmetric monoidal under $\sqcup$.
Theorem 3.1. $\mathrm{Fin}^{\simeq}$ is the free $E_\infty$-space on one point. For every $E_\infty$-space $M$, evaluation at a singleton is an equivalence $\mathrm{Map}_{E_\infty}(\mathrm{Fin}^{\simeq},M)\simeq M$.
*Proof.* The free $E_\infty$-space on $X$ is $\coprod_n(X^n)_{h\Sigma_n}$ [4, §3.1.3]. For $X=*$ this is $\coprod_nB\Sigma_n$, with multiplication induced by block sum $\Sigma_m\times\Sigma_n\to\Sigma_{m+n}$, i.e. by disjoint union. The equivalence is the universal property of free algebras. $\square$
Definition 3.2 (Utilitarian valuation). Let $M$ be an $E_\infty$-space of influence values. A utilitarian valuation is a symmetric monoidal functor $\iota:\mathrm{Fin}^{\simeq}\to M$. The influence of $p$ in a polity is $\iota(S_p)$.
Functoriality on bijections is anonymity: relabelling persons cannot change influence, as in May's theorem [7]. Monoidality is additivity over disjoint groups of participants.
Corollary 3.3. If $M$ is discrete, every utilitarian valuation is $\iota(S)=|S|\cdot m$ with $m=\iota(\{*\})$. For $M=(\mathbb R,+)$, influence is proportional to $N_p$.
*Proof.* For discrete $M$, maps factor through $\pi_0\mathrm{Fin}^{\simeq}=\mathbb N$, and a monoid map $\mathbb N\to M$ is determined by the image of $1$. $\square$
Thus the count in (T) is not an additional assumption: anonymity plus additivity force it, as in equal-weight utilitarian aggregation [8]. If $M$ is not discrete, $\iota$ is still determined by $m$, but carries symmetry data. In the Picard groupoid of $\mathbb Z/2$-graded lines with $m$ odd, $\iota(S)=m^{\otimes|S|}$ and $\Sigma_n$ acts by the sign. Such data is invisible to every numerical index.
Proposition 3.4 (Payments). Let $\Phi$ assign rationals to $\pi$-finite spaces. Assume it is equivalence invariant and additive, with $\Phi(*)=1$, and $\Phi(E)=\Phi(F)\Phi(B)$ for every fibre sequence $F\to E\to B$ with $B$ connected. Then $\Phi(X)=|X|$.
*Proof.* Additivity gives $\Phi(A)=|A|$ for finite sets $A$ and reduces to connected $X$. The sequence $K(A,n-1)\to*\to K(A,n)$ gives $\Phi(K(A,n))=\Phi(K(A,n-1))^{-1}$, hence $|A|^{(-1)^n}$; the case $n=1$ allows nonabelian $A$. For connected $X$, the Postnikov sequences $K(\pi_n,n)\to X_{\le n}\to X_{\le n-1}$ have connected bases and finitely many nontrivial stages. Induction yields the product formula. $\square$
Homotopy cardinality satisfies these axioms [1,3]. Payment-weighted influence $R_p$ is therefore the unique count additive over disjoint payments and multiplicative over cost-sharing fibrations.
4. Opinion transport
Let $x\in\mathbb R^U$ be opinions, $a_p\in\mathbb R$ the position advocated by the operator of $p$, and $\alpha\in(0,1]$ an attention rate. One round of exposure is the DeGroot-type step [6] $$ x'_u=x_u+\frac{\alpha}{|\Pi_u|}\sum_{p\in\Pi_u}(a_p-x_u)\quad(\Pi_u\neq\varnothing),\qquad x'_u=x_u\quad(\Pi_u=\varnothing). $$ Persons divide attention equally among platforms they pay for. The mean shift is $\Delta=\sum_p\Delta_p$, where $$ \Delta_p=\frac{\alpha}{|U|}\sum_{u\in S_p}\frac{a_p-x_u}{|\Pi_u|}. $$ $\Delta_p$ depends on $E$ only through its $\mathbb B$-decategorification. (T) asserts that it factors further through the column sums $N_p$.
Proposition 4.1. Suppose the polity is single-homing ($|\Pi_u|\le1$) and has no selection: $\frac1{N_p}\sum_{u\in S_p}x_u=\bar x$, the population mean, whenever $N_p>0$. Then $$ \Delta_p=\alpha\,\frac{N_p}{|U|}\,(a_p-\bar x). $$ The operator's opinion share is $N_p/|U|$, a utilitarian valuation with $m=\alpha/|U|$.
*Proof.* Substitute $|\Pi_u|=1$ and the participant mean. $\square$
Proposition 4.2 (Both hypotheses are needed). (a) Multi-homing. Let $U=\{1,2,3,4\}$ and $\Pi=\{p,q,r\}$. Polity A has $S_p=\{1,2\}$, $S_q=\{2,3\}$, $S_r=\{3,4\}$. Polity B has $S_p=S_q=\{1,2\}$, $S_r=\{3,4\}$. Both have $N=(2,2,2)$. For constant opinions $x_u=\bar x$, $\Delta_p=\frac{\alpha}{|U|}\tilde N_p(a_p-\bar x)$ with fractional count $\tilde N_p=\sum_{u\in S_p}|\Pi_u|^{-1}$. A gives $\tilde N=(\tfrac32,1,\tfrac32)$; B gives $(1,1,2)$. (b) Selection. Let $U=\{1,2\}$, $x=(0,1)$, $a_p=1$. The single-homing polities $S_p=\{1\}$ and $S_p=\{2\}$ both have $N_p=1$, with $\Delta_p=\alpha/2$ and $\Delta_p=0$.
*Proof.* Direct computation, also checked by script. $\square$
Always $\sum_p\tilde N_p=|\bigcup_pS_p|$: attention shares redistribute persons reached, but not in proportion to $N$.
5. Coalitions and descent
For a bloc $Q\subseteq\Pi$, for example platforms with a common owner, put $S_Q=\bigcup_{p\in Q}S_p$. Let $C_\bullet$ be the Čech nerve of $\coprod_{p\in Q}S_p\to S_Q$, so that $C_n=\coprod_{(p_0,\dots,p_n)\in Q^{n+1}}S_{p_0}\cap\dots\cap S_{p_n}$.
Theorem 5.1. (a) The realization of $C_\bullet$ in $\mathcal S$ is $S_Q$. (b) $N_Q=\sum_{\varnothing\neq R\subseteq Q}(-1)^{|R|+1}\,\big|\bigcap_{r\in R}S_r\big|$. (c) $N_Q\le\sum_{p\in Q}N_p$, with equality iff the $S_p$ are pairwise disjoint.
*Proof.* (a) The map is surjective on $\pi_0$, hence an effective epimorphism, and the Čech nerve of an effective epimorphism in an $\infty$-topos realizes to its target [5, §6.2.3]. Concretely, over $u\in S_Q$ the nerve is that of the indiscrete groupoid on $Q_u=\{p\in Q:u\in S_p\}\neq\varnothing$, which is contractible. (b) The right side equals $\sum_{u\in S_Q}\sum_{\varnothing\neq R\subseteq Q_u}(-1)^{|R|+1}=\sum_{u\in S_Q}\big(1-(1-1)^{|Q_u|}\big)=N_Q$. Each inner sum is the Euler characteristic $1$ of the simplex on $Q_u$. (c) Union bound. $\square$
Bloc influence is computed by gluing. The obstruction to additivity is the overlap data: double intersections are 1-simplices, triple intersections are 2-simplices, and so on. Applying (T) platform by platform overcounts multi-homing persons. For disjoint jurisdictions $U=\coprod_jU_j$, counts are additive: $N_p=\sum_jN_p^{(j)}$.
6. Transmission: why person counts do not compose
A reflexive social relation is a morphism $G:U\to U$ in $\mathbf{Pol}$ containing the diagonal. Here $G_{v,u}\neq\varnothing$ means $v$ sees what $u$ sees. By (2.1), $(E\circ G)_{v,p}\neq\varnothing$ iff some contact $u$ of $v$ participates in $p$. The reach of $p$ is $\rho_p=|\{v:(E\circ G)_{v,p}\neq\varnothing\}|$.
Proposition 6.1. (a) If $G$ is the diagonal then $\rho_p=N_p$. (b) $\rho_p=N_p$ for every polity on $U$ iff $G_{v,u}=\varnothing$ for all $v\neq u$. (c) In general $\rho_p$ is not a function of $N_p$. On the reflexive path $1-2-3-4$, $S_p=\{1,2\}$ gives $\rho_p=3$, while $S_q=\{1,4\}$ gives $\rho_q=4$. (d) Exposures counted with multiplicity do compose: $X_p:=\sum_v|(E\circ G)_{v,p}|=\sum_u k_u|E_{u,p}|$ with $k_u=\sum_v|G_{v,u}|$. If $k_u=k$ for all $u$, then $X_p=kR_p$.
*Proof.* (a) is (2.1). (b) If $G_{v,u}\neq\varnothing$ with $v\neq u$, then $S_p=\{u\}$ gives $\rho_p\ge2>N_p$; the converse is (a). (c) Closed neighbourhoods $\{1,2\}\cup\{1,2,3\}$ versus $\{1,2\}\cup\{3,4\}$. (d) Functoriality of $|{-}|$ (Proposition 2.5). $\square$
The asymmetry is algebraic. Composition is computed in $\mathbb Q_{\ge0}$ or $\mathbb B$. Payments and exposures are linear in $\mathbb Q_{\ge0}$, so a version of (T) survives for exposures in regular networks. Person counts are column sums of $\mathbb B$-matrices, and column sums do not respect $\mathbb B$-multiplication. Network effects, the defining feature of platforms, thus leave the domain of Theorem 3.1. The relevant object is a category of graphs, which has no universal property forcing influence to be a count.
7. Summary
| Hypothesis | Makes (T) hold via | Failure when dropped | |---|---|---| | Anonymity and additivity | Theorem 3.1, Corollary 3.3 | non-discrete values carry symmetry data | | Fixed truncation level | Proposition 2.3 | persons, memberships, payments diverge | | Single-homing, no selection | Proposition 4.1 | Proposition 4.2 | | Disjoint blocs | Theorem 5.1 | overlap (Čech) terms | | No transmission | Proposition 6.1(a,b) | reach is not a count |
(T) is exact on the free part of a polity. Elsewhere $N_p$ is the image of the polity under a non-functorial map $\mathbf{Pol}\to\mathbb N^{\Pi}$. A faithful invariant retains at least the $\mathbb B$-, $\mathbb N$- and $\mathbb Q_{\ge0}$-decategorifications of $E$ together with $G$.
8. Limitations
The model is static and linear, and opinions are real numbers. Operators are identified with their platforms. Unpaid participation, for example advertising-funded users, needs a second span; totals are coproducts in $\mathbf{Pol}$. We use no data and make no claim about the magnitude of real influence. We do not evaluate utilitarianism normatively. Theorem 3.1 and the existence half of Proposition 3.4 rely on [1,3,4]; Theorem 5.1(a) relies on [5]. Influence maximization in social networks [10] and multi-homing in two-sided markets [11] are related but not formalized here.
Disclosure of automation. Anonymous. An AI assistant drafted the formalization, proofs and text at the researcher's request, and the researcher authorized submission. The examples in Propositions 4.2 and 6.1 and the identities in Theorem 5.1(b) were checked by script on random instances. No proof-assistant certification or independent human verification is asserted. The journal determines review status.
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