I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions:

1.) The naive approach to define a homotopy would be ('naive' in my opinion of course) the following:

Let $(L,(l)_{k\in\mathbb{N}})$ and $(M,(d)_{k\in\mathbb{N}})$ be two Lie infinity algebras, let $f_\infty,g_\infty:L \to M$ be two morphism (in the most general sense) and let $(C(L),Q_L)$ and $(C(M),Q_M)$ be the appropriate differential graded coalgebras with induced morphism $F,G:C(L)\to C(M)$.

Then a **homotopy** between $F$ and $G$ is a degree $+1$ map

$H:C(L)\to C(M)$ such that $F-G = HQ_L \pm Q_MH$

let signs and additional structure of $H$ (linear, coalgebra, ...) aside for a moment.

It is strange, however, that I never saw this approach in the literature. Is this definition of homotopy equivalent to the previously mentioned approaches in question I?

Now for what is more important:

2.) The homotopy theory of Lie infinity algebras as given by Urs Schreiber in question I is obtained by 'transferring' the homotopy theory of differential graded Lie algebras 'along' an adjunction

$F\dashv G$

($F:L_\infty Alg \to DGLA$ and $G:DGLA \to L_\infty Alg$)

Now the question that really irritates me for quite some time is: how can we be sure that we get the correct homotopy theory of Lie infinity algebras by transferring its 'shadow' in the category of DG Lie algebra back along the previous mentioned adjunction? To me it looks like we can not rule out that there is a more general definition of weak equivalences in the category of Lie infinity algebras, which just project under $F$ onto those we already know.

Sorry if the second question is vague.

Edit; I used the 'Lie algebra cohomology tag' since the homotopy theory of Lie infinity algebras affects Lie algebra cohomology, too.