E0055 proposition

Cofree coalgebra lift

The symmetric coalgebra \((\SS_+(V), \bar{\Delta})\) E0052 is the cofree conilpotent cocommutative coalgebra on \(V\): every linear map into \(V\) lifts uniquely to a coalgebra morphism.

(1) Proposition. Let \((C, \Delta_C)\) be a conilpotent coalgebra and \(f: C \to V\) a linear map. There is a unique coalgebra morphism \(f^+: C \to \SS(V)\) with \(\pi_1 \circ f^+ = f\), given by the convolution exponential:

\[ f^+ = \exp_*(f) = \sum_{k \geq 0} \frac{1}{k!} f^{*k}. \]

The sum terminates on each element of \(C\) by conilpotence.

Proof. \(f: C \to V = \SS^1(V)\) is primitive in \(\Hom(C, \SS(V))\) with the convolution product \(f * g = \mu \circ (f \otimes g) \circ \Delta_C\). By the exp/log correspondence E0054 for convolution algebras, \(\exp_*(f)\) is group-like in \(\Hom(C, \SS(V))\), i.e. a coalgebra morphism. Conversely, any coalgebra morphism \(g\) has \(\log_*(g) = \pi_1 \circ g = f\), giving uniqueness.

(2) Corollary. The symmetric pushforward \(D_+(\phi; x)\) E0027 is the unique coalgebra morphism \(ST_k(X; x) \to ST_k(Y; y)\) whose degree-one component is the differential \(D(\phi; x)\). Its partition formula is the convolution exponential of the differential.

Proof. \(ST_k(X; x)\) with the reduced shuffle coproduct E0052 is conilpotent (truncated at degree \(k\)). The differential \(D(\phi; x): ST_k(X; x) \to T_y Y \subseteq ST_k(Y; y)\) is a linear map into the degree-one part. By the proposition above, \(D(\phi; x)^+ = \exp_*(D(\phi; x))\) is the unique coalgebra morphism extending it. Expanding the convolution exponential: \(\exp_*(D(\phi; x))(v_1 \cdots v_r) = \sum_{m=1}^r \frac{1}{m!} (D(\phi; x))^{*m}(v_1 \cdots v_r) = \sum_{\pi \in \Part(r)} \prod_{A \in \pi} D(\phi; x; v_A)\), which is the partition formula defining \(D_+\) E0027.