$$
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% ============================================
% CHAPTER 1: DIFFERENTIAL DUALITY
% ============================================
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% ============================================
% CHAPTER 2: DISCRETE DUALITY
% ============================================
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$$
Discrete functoriality and adjunction
The cubical pushforward \(\Delta_+\) E0016 is an exact functor,
and the pullback \(\Delta^+\) is its contravariant adjoint.
Throughout, \(y = \phi(x)\) and \(z = \psi(y)\); on cube measures,
\(\phi_*\) is the linear extension of \(\delta(p) \mapsto
\delta(\phi(p))\), and \(\phi^* f = f \circ \phi\) is the pullback
of functions.
(1) Theorem (Exact functoriality). For arbitrary maps
\(\phi: X \to Y\), \(\psi: Y \to Z\) and every \(k \geq 1\),
\[
\Delta_+(\psi \circ \phi; x)
= \Delta_+(\psi; y) \circ \Delta_+(\phi; x),
\qquad
\Delta_+(\id_X; x) = \id.
\]
Proof.
The geometric pushforward composes vertexwise:
\((\psi \circ \phi)_* = \psi_* \circ \phi_*\). Insert the
identity \(\zeta_y \circ \mu_y = \id\) (Möbius–zeta inversion
E0002) between the two factors in
\(\mu_z \circ \psi_* \circ \phi_* \circ \zeta_x =
(\mu_z \circ \psi_* \circ \zeta_y)
\circ (\mu_y \circ \phi_* \circ \zeta_x)
= \Delta_+(\psi; y) \circ \Delta_+(\phi; x)\).
For the identity map,
\(\Delta_+(\id_X; x) = \mu_x \circ (\id_X)_* \circ \zeta_x
= \mu_x \circ \zeta_x = \id\).
(2) Theorem (Discrete adjunction). For arbitrary
\(\phi: X \to Y\), \(f: Y \to G\), and \(c \in CT_k(X; x)\),
\[
\phi_*\, \delta(x; c) = \delta(y; \Delta_+(\phi; x)\, c),
\qquad
\Delta(\phi^* f; x; c) = \Delta(f; y; \Delta_+(\phi; x)\, c).
\]
Proof.
By Möbius–zeta inversion E0002,
\(\zeta_y(\Delta_+(\phi; x) c) = \phi_*(\zeta_x c)\): the
geometric realization of the pushed-forward tangent cube is
the vertexwise image of the original geometric cube. The
first identity follows vertexwise on cube measures; the
second by pairing against \(f\).
Proof.
Precomposition reverses composition; apply exact
functoriality.
(4) Validation (AI review, 2026-07-19, claude-fable-5, pass).
All three proofs verified against E0016 definitions, with
basepoint tracking \(\phi_*: \Cube_k(X;x) \to \Cube_k(Y;y)\) and
termwise agreement over \(T \subseteq [k]\) in the adjunction;
edge cases \(k = 0\), identity map, constant \(\phi\) checked.
Fixed: proof of \(\Delta_+(\id_X;x) = \id\) was missing (added
one line); intro now declares \(y, z\) and imports \(\phi_*\) on
measures and \(\phi^* f\), which were previously undefined.