Shifted coalgebra definition of an L∞-algebra

Let $k$ be a characteristic-zero field (Field, The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise), and let $V={⨁}_{j\in \mathbb{Z}}{V}^j$ be a graded $k$-vector space. Use the cohomological suspension

$W=V[1],W^j=V^{j+1}\mathrm{.}$

The reduced cofree conilpotent cocommutative coalgebra on $W$ is the $S^c(W)={⨁}_{n\ge 1}{S}^n(W)$ of Reduced graded symmetric coalgebra with its shuffle coproduct, with its cofree property proved in The reduced graded symmetric coalgebra is cofree conilpotent cocommutative.

An L∞-algebra on $V$ in the shifted coalgebra convention is a degree $+1$ coderivation $Q:S^c(W)\to S^c(W)$ with $Q^2=0$ and no arity-zero component. A coderivation satisfies $\bar{\mathrm{\Delta }}Q=(Q\otimes 1+1\otimes Q)\bar{\mathrm{\Delta }}$, with the Koszul sign on $1\otimes Q$, as defined in Coderivations of the reduced graded symmetric coalgebra are determined by corestrictions. Its corestrictions are

$q_n={\pi }_{1}Q{\mathrm{\mid }}_{S^n(W)}:S^n(W)⟶W,\mathrm{\mid }{q}_n\mathrm{\mid }=+1(n\ge 1),$

where ${\pi }_1:S^c(W)\to S^1(W)=W$ is the length-one projection. Conversely, each such family determines a unique coderivation by that lemma. The condition $Q^2=0$ is equivalent to the higher Jacobi identities proved in Square-zero coderivations are equivalent to the shifted higher Jacobi identities ↗.

This is the shifted convention only: no unsuspended skew operations $l_n$ or translation formula for them is asserted. Curved $L_{\mathrm{\infty }}$ structures with an arity-zero term are outside this definition.