# Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

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?

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This doesn't answer your question directly, but there's a picture where instead of looking at intertwining maps $V\to W$ as your morphisms, you look at maps from your operad to $End(V,W)$ viewed as a bimodule over $End(V)$ and $End(W)$. I think this gives the right picture whatever cofibrant replacement you use. – Gabriel C. Drummond-Cole Jul 17 '13 at 20:39
Can you make this a precise math question? For example, will you be satisfied if given an arbitrary replacement, you can find the homotopy category of the nonlinear morphisms"? I suspect not, but it's not clear to me what exactly you want. – Joey Hirsh Jul 17 '13 at 21:28
@DavidWhite: Yes, I've talked to Bruno here about various things. I hadn't seen that question, though! – Theo Johnson-Freyd Jul 17 '13 at 23:45
Hi @JoeyHirsh, I would accept an answer that described "the homotopy category of nonlinear morphisms" given some arbitrary replacement. I'm hoping, though, for something that would give you "nonlinear morphisms" for other (pr)operads. Recently I've been thinking a lot about the properad of Lie bialgebras (you and I talked about this a bit in March), which is Koszul, and I can compute a cofibrant replacement using this, but because there's no "free" functor for representations of a properad, I don't see how to decide what the "nonlinear morphisms" should be. – Theo Johnson-Freyd Jul 17 '13 at 23:49
there's no free functor for algebras over a properad. there is however a free P-module functor, and I believe this allows for a construction which is similar to the bar construction for algebras, although the output is not a coalgebra of dual type, it is some other thing, maybe a comodule, of dual type. I believe this paper arxiv.org/abs/0807.1241 provides an example of what I'm talking about. – Joey Hirsh Jul 18 '13 at 14:26