E0028 theorem notation E0020 → E0001

Smooth functoriality and adjunction

The symmetric pushforward \(D_+\) E0027 is functorial, and the smooth adjunction intertwines pullback of functions with pushforward of tangent probes.

(1) Theorem (Coalgebra property). \(D_+(\phi; x)\) is a coalgebra morphism for the shuffle coproduct:

\[ \Delta^\times \circ D_+(\phi; x) = (D_+(\phi; x) \otimes D_+(\phi; x)) \circ \Delta^\times. \]

Proof. Evaluate on \(v_1 \cdots v_r\) and set \(w_A := D(\phi; x; v_A)\). Applying \(\Delta^\times\) to each partition term splits its block set into two parts: the data \((\pi, \pi_1, \pi_2)\) with \(\pi = \pi_1 \sqcup \pi_2\) correspond bijectively to \((I \sqcup J = [r], \pi_1 \in \Part(I), \pi_2 \in \Part(J))\), giving \((D_+ \otimes D_+) \Delta^\times(v_1 \cdots v_r)\).

(2) Corollary (Coalgebra lift). \(D_+(\phi; x)\) is the unique coalgebra morphism \(ST_k(X; x) \to ST_k(Y; y)\) whose degree-one component is the differential \(D(\phi; x)\).

Proof. The coalgebra property is the theorem above. Uniqueness: by the cofree property of the truncated symmetric coalgebra E0055, a coalgebra morphism \(ST_k \to ST_k\) is determined by its degree-one projection, which is the differential \(D(\phi; x)\). Its partition expansion is the definition of \(D_+\) E0027.

(3) Theorem (Smooth functoriality). For \(\phi\) \(C^k\) near \(x\) and \(\psi\) \(C^k\) near \(y\),

\[ D_+(\psi \circ \phi; x) = D_+(\psi; y) \circ D_+(\phi; x). \]

Proof. Both sides are coalgebra morphisms \(ST_k(X; x) \to ST_k(Z; z)\) with the same degree-one component: \(D(\psi \circ \phi; x; v) = D(\psi; y; D(\phi; x; v))\) is the first-order chain rule. By the coalgebra lift, they are equal.

(4) Theorem (Smooth adjunction). For \(\phi\) \(C^k\) near \(x\), \(f \in C^k(Y, G)\) near \(y\), and \(\xi \in ST_k(X; x)\),

\[ D(\phi^* f; x; \xi) = D(f; y; D_+(\phi; x)\, \xi). \]

Proof. Both sides are linear in \(\xi\) and agree at \(\xi = 1\) (both give \(f(y)\)). On monomials \(\xi = v_1 \cdots v_r\), the left side is \(D^r(f \circ \phi; x; v_1, \dots, v_r)\). Expanding by the partition Faà di Bruno E0022: \(D^r(f \circ \phi; x; v_\bullet) = \sum_{\pi \in \Part(r)} D^{|\pi|}(f; y; (D(\phi; x; v_A))_{A \in \pi}) = D(f; y; D_+(\phi; x; v_1 \cdots v_r))\).

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