E0027 definition notation E0020 → E0001

Symmetric pushforward and pullback

The symmetric tangent and cotangent spaces host the smooth counterpart of the cubical pushforward E0016. The symmetric pushforward \(D_+\) is defined by the partition formula and is the unique coalgebra morphism determined by the differential.

(1) Definition (Symmetric tangent spaces).

  • The symmetric tangent space of order \(k\) at \(x\) is \(ST_k(X; x) := \bigoplus_{r=0}^k \SYM^r(E)\), where \(\SYM^r(E)\) is the \(r\)-th symmetric power of the model space \(E\).
  • Elements are symmetric tensors \(\xi = s \cdot 1 + v_1 + v_1 v_2 + \cdots\), where \(v_1 \cdots v_r\) denotes the symmetric product of \(r\) vectors.
  • The shuffle coproduct is \(\Delta^\times(v_1 \cdots v_r) = \sum_{I \sqcup J = [r]} v_I \otimes v_J\), making \(ST_k(X; x)\) a cocommutative coalgebra.
  • The differential pairing of \(f: X \to G\) at \(x\) with \(\xi \in ST_k(X; x)\) is \(D(f; x; \xi)\), defined on monomials by \(D(f; x; v_1 \cdots v_r) := D^r(f; x; v_1, \dots, v_r)\) and \(D(f; x; 1) := f(x)\).

(2) Definition (Differential and symmetric pushforward).

  • The differential of \(\phi: X \to Y\) (\(C^k\) near \(x\)) is the linear map \(D(\phi; x): ST_k(X; x) \to T_y Y\) defined by

    \[ D(\phi; x; v_1 \cdots v_r) := D^r(\phi; x; v_1, \dots, v_r), \qquad 1 \leq r \leq k, \]

    with \(D(\phi; x; 1) := 0\). It collects the derivatives of all orders into a single linear map landing in degree one on the target.

  • The symmetric pushforward \(D_+(\phi; x): ST_k(X; x) \to ST_k(Y; y)\) is the linear map with \(D_+(\phi; x; 1) := 1\) and

    \[ D_+(\phi; x; v_1 \cdots v_r) := \sum_{\pi \in \Part(r)} \prod_{A \in \pi} D(\phi; x; v_A), \qquad 1 \leq r \leq k. \]

    The right side is symmetric and multilinear in \((v_1, \dots, v_r)\).

(3) Definition (Symmetric pullback).

  • The symmetric pullback of \(\phi: X \to Y\) (\(C^k\) near \(x\)) is the map \(D^+(\phi; x): ST^k(Y; y) \to ST^k(X; x)\) defined by adjunction:

    \[ \langle D^+(\phi; x)\, \omega, \xi \rangle = \langle \omega, D_+(\phi; x)\, \xi \rangle. \]

  • The smooth adjunction states that the differential pairing intertwines pullback and pushforward: for \(f \in C^k(Y, G)\),

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

  • \(D^+(\phi; x)\) is an algebra morphism for the pointwise product on \(ST^k\).

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