A $L_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative coalgebra, cofree in the category of *locally nilpotent* differential graded coalgebras and their morphisms are coalgebra
morphisms that commute with the codifferential.

Breaking this compact definition down into something more concrete, the category of $L_\infty$-algebras can equally be defined in the following way:

A **$L_\infty$-algebra** is a $\mathbb{Z}$-graded vector space $V$, together with a sequence of graded anti-symmetric, $k$-linear maps

$D_k:V \times \cdots \times V \to V$,

homogeneous of degree $-1$,such that the 'weak' Jacobi identity

$ \sum_{p+q=n+1}\sum_{\sigma \in Sh(q,n-q)}\epsilon(\sigma;x_1,\ldots,x_n) D_p(D_q(x_{\sigma(1)},\ldots,x_{\sigma(q)}),x_{\sigma(q+1)},\ldots,x_{\sigma(n)})=0 $

is satisfied, for any $n\in\mathbb{N}$. Where $\epsilon$ is the Koszul sign and $Sh(p,q)$ is the set of suffle permutations.

A **morphism** of $L_\infty$-algebras $(V,D_{k\in\mathbb{N}})$
and $(W,l_{k\in\mathbb{N}})$ is a sequence $f_{k\in\mathbb{N}}$ of
graded-antisymmetric, $k$-linear maps

$ f_k : V\times \cdots \times V \to W $

homogeneous of degree zero, such that the equation

$ \sum_{p+q=n+1}\sum_{\sigma \in Sh(q,n-q)}\epsilon(\sigma;x_1,\ldots,x_n) f_p(D_q(x_{\sigma(1)},\ldots,x_{\sigma(q)}),x_{\sigma(q+1)},\ldots,x_{\sigma(n)})=\\ \sum_{k_1+\cdots+k_j=n}^{k_i\geq 1}\sum_{\sigma \in Sh(k_1,\ldots,k_j)} \epsilon(\sigma;x_1,\ldots,x_n) l_j(f_{k_1}(x_{\sigma(1)},\ldots,x_{\sigma(k_1)}),\ldots, f_{k_j}(x_{\sigma(n-k_j+1)},\ldots,x_{\sigma(n)})) $

is satisfied, for any $n\in\mathbb{N}$.

This defines the category of $L_\infty$-algebras, sometimes called the category of $L_\infty$-algebras with **weak morphisms**.

Now after that long and tedious definition, the question is:

What is a reasonable definition of a **homotopy** between two (weak) morphisms
$f:V\to W$ and $g:V\to W$ of $L_\infty$-algebras? (And why?)

Edit: A lot of information pointing towards a definition of such a homotopy (or 2-morphism in $(\infty,1)$-categorical language) is spread out in the net. Much on the $n$-category cafe, like in https://golem.ph.utexas.edu/category/2007/02/higher_morphisms_of_lie_nalgeb.html and in the nLab. However it looks like an explicit equation still isn't available.

I would do the tedious calculations myself, since I can get a lot of joy out of
such huge and delicate computations, but I'm unable to finde a calculable way to
achive that goal. (Such a way should have the potential to apply to the higher homotopies too, hopefully leading towards an explicit description of the hom-**space** in this category)

P.S.: The tags are not very well suited, feel free to change them