Recall that a *Lie algebra* is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a model category on the category of dg operads over a field $\mathbb K$ of characteristic $0$, for which the weak equivalences are the homomorphisms that are isomorphisms on homology, and the fibrations are surjective homomorphisms. See Loday and Vallette, *Algebraic Operads*.

For this model category structure, $\mathrm{Lie}$ is not cofibrant, but it has, of course, cofibrant replacements. One such cofibrant replacement is the operad $L_\infty$, given by Koszul duality ($\mathrm{Lie}$ is Koszul). A short way of defining $L_\infty$ is that a module $V$ for $L_\infty$ (also called a *flat $L_\infty$ algebra*) is as follows. Let $\operatorname s$ denote the shift operation (and I'm not going to try to figure out whether it shifts up or down), and $\operatorname{Sym}$ the symmetric algebra construction (with the usual sign rules). Then $\operatorname{Sym}(\operatorname s V)$ is a dg Hopf algebra for any vector space $V$ (using the isomorphism $\operatorname{Sym}(X\oplus Y) \cong \operatorname{Sym}(X)\otimes \operatorname{Sym}(Y)$ and the diagonal map $X \to X\oplus X$ and functoriality). Then an action of $L_\infty$ on $V$ is the same as a degree-$(-1)$ coderivation $\delta$ on $\operatorname{Sym}(\operatorname s V)$ that vanishes on $\operatorname{Sym}^{\leq 1}(V) = V \oplus \mathbb K$ and satisfies the Maurerâ€“Cartan equation $(\partial + \delta)^2 = 0$. Since any coderivation on a symmetric algebra is determined by its projection to $\operatorname{Sym}^1$, $\delta$ is the same data as a sequence of maps $\bigwedge^n V \to V$ of degree $n-2$, for $n\geq 2$. Note that the differential $\partial +\delta$ makes $\operatorname{Sym}(\operatorname s V)$ into a dg coalgebra, but not a dg Hopf algebra.

If you think of $L_\infty$ as an operad, then there is a natural notion of *(linear) homomorphism* of $L_\infty$ algebras $V,W$, namely a chain complex map $V \to W$ that for each $n$ intertwines the map $\bigwedge^n V \to V$ with the map $\bigwedge^n W \to W$. But by thinking about $L_\infty$ algebras in terms of dg coalgebras, one can also define a *nonlinear homomorphism* $V \to W$ to be any homomorphism of dg coalgebras $\operatorname{Sym}(\operatorname s V) \to \operatorname{Sym}(\operatorname s W)$. This nonlinear notion tends to be the correct one in many parts of mathematics.

Question:If I didn't know about this coalgebra picture of $L_\infty$ algebras, and just had some random cofibrant replacement $\mathrm{Lie}_\infty$ of $\mathrm{Lie}$, could I have invented, say from the theory of algebraic model categories, some sort of "nonlinear homomorphism"? For linear homomorphisms, I believe that different models of $\mathrm{Lie}_\infty$ have Quillen-equivalent representation theories; is this still true for the nonlinear homomorphisms?