E0001 notation ✓ formal

Cubical setting on abelian groups

Standing notation for the discrete Möbius calculus. Statement nodes import this environment; no result is asserted here.

(1) Notation (Sets, cubes, grids).

  • \([n] = \set{1, \dots, n}\) is the standard index set.
  • \(\KP(S)\) is the power set of a finite set \(S\).
  • \(\KP_+(S) = \KP(S) \setminus \set{\emptyset}\) is the set of nonempty subsets; \(\KP(n) = \KP([n])\).
  • A cube in \(X\) is a map \(a: \KP(S) \to X\); it has \(2^{|S|}\) vertices \(a(T)\) and legs \(a(\set{i})\).
  • A grid in \(X\) is a map \(A: \IN_0^S \to X\); the points \(A(1_T)\) for \(T \subseteq S\) form its coordinate cube.

(2) Notation (Multi-indices).

  • A multi-index on a finite set \(S\) is \(\alpha \in \IN_0^S\), with componentwise partial order \(\beta \leq \alpha\).
  • The weight is \(|\alpha| = \mathrm{wt}(\alpha) = \sum_s \alpha_s\).
  • The height is \(\mathrm{ht}(\alpha) = \max_s \alpha_s\).
  • The factorial is \(\alpha! = \prod_s \alpha_s!\).
  • The falling factorial is \((\alpha)_\beta = \prod_s (\alpha_s)_{\beta_s}\), where \((n)_r = n(n-1)\cdots(n-r+1)\).
  • The Boolean realization of \(\alpha \in \IN_0^S\) is the finite set \(S(\alpha) = \set{(s,i) : s \in S,\ 0 < i \leq \alpha_s}\), with projection \(\pi: S(\alpha) \to S\).
  • The fiber measure of a map \(q: S' \to S\) of finite sets is \(\nu(q) \in \IN_0^S\), \(\nu(q)_s = |q^{-1}(s)|\); in particular \(\nu(\pi) = \alpha\).
  • \(\KP(S) \hookrightarrow \IN_0^S\) via \(U \mapsto 1_U\); the image consists of the multi-indices of height \(\leq 1\).

(3) Notation (Higher power sets and coverings).

  • The higher power sets are defined recursively: \(\KP_+^0(S) = S\), \(\KP_+^1(S) = \KP_+(S)\), and \(\KP_+^r(S) := \KP_+(\KP_+^{r-1}(S))\) for \(r \geq 2\).
  • The leaf support \(\lf(K) \subseteq S\) for \(K \in \KP_+^r(S)\) is defined recursively: \(\lf(K) := K\) for \(r = 1\) and \(\lf(K) := \bigcup_{L \in K} \lf(L)\) for \(r \geq 2\).
  • The set of \(r\)-fold coverings, for \(r \geq 1\), is \(\Cov_r(S) := \set{K \in \KP_+^r(S) \mid \lf(K) = S}\).
  • For \(S = [k]\), we write \(\KP_+^r(k)\) and \(\Cov_r(k)\).
  • \(\Cov(S) := \Cov_2(S)\) is the set of coverings of \(S\).

(4) Convention (Maps and points).

  • \(X, Y, Z\) denote abelian groups; points are \(x \in X\), \(y \in Y\), \(z \in Z\).
  • \(g: X \to Y\) and \(f: Y \to Z\) are arbitrary maps (no linearity, continuity, or regularity assumed), with \(g(x) = y\), \(f(y) = z\).
  • Directions are \(u_1, \dots, u_k \in X\); formulas evaluate maps only at points \(x + \sum_{i \in S} u_i\), so domains may be commutative monoids where noted.

(5) Validation (AI review, 2026-07-19, claude-fable-5, pass). Checked every bullet for internal consistency and the in-line claims (\(\nu(\pi) = \alpha\), image of \(U \mapsto 1_U\) = height \(\leq 1\), \(2^{|S|}\) vertices); edge cases \(S = \emptyset\) (\(\KP_+^r(\emptyset) = \Cov_r(\emptyset) = \emptyset\)), \(r = 1\) boundary of both recursions, \(\alpha = 0\); spot-checked symbol usage in E0006–E0008, E0011. Two minor fixes: restricted \(\Cov_r\) to \(r \geq 1\) (leaf support is undefined at \(r = 0\)) and defined \(\Cov(S)\) for general \(S\) (E0006 uses it).

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