lin-0281
Local-to-global, causal, and decision notation
Notation Meaning Role in lincs \(\mathcal F(U)\) sections over context \(U\) Local predictions, arguments, evidence, or model fragments. \(\rho ^U_V\) restriction map Transports a section on \(U\) to \(V\subseteq U\). \(\check C^\bullet (\mathcal U,\mathcal F)\) Cech complex of a cover Records overlap compatibility and higher gluing information. \(\delta _{\check C}\) Cech coboundary Measures failure of local data to agree on overlaps. \(\operatorname {colim}\), \(\lim \) colimit and limit Universal aggregation and compatibility constructions. \(P(Y\mid X=x)\) observational conditional Evidence obtained by conditioning, without an intervention claim. \(P(Y\mid \operatorname {do}(X=x))\) interventional distribution A measure defined under a declared intervention mechanism. \(D_{\mathrm{KL}}(P\Vert Q)\) Kullback–Leibler divergence Compares identified measures but does not certify causal identification. \(V^\pi ,Q^\pi \) value functions Decision quantities under policy \(\pi \). \(T^\pi \), \(T^\ast \) Bellman operators Policy and optimality factorizations used in decision and RL chapters. \(\operatorname {Ran}_J(J^\ast \operatorname {Lan}_J F)\) universal decision composite Extension followed by consistency relative to the information made visible by \(J\).
A symbol does not acquire causal, decision, or repair semantics from its shape alone. Its type, declaration, observer, and admission contract determine what the notation licenses in a particular application.