TITLE: What is a homotopy between $L_\infty$-algebra morphisms
QUESTION [18 upvotes]: 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

REPLY [2 votes]: There is a plethora of model structures for L-infinity algebras (going back to Quillen of course, but notably described and related in the great article by Jonathan Pridham arXiv:0705.0344). Also structures of categories of fibrant objects. Each of these induces a model for homotopies of 1-morphisms of $L_\infty$-algebras, for instance a right homotopy given by mapping into a path space object. What these are can be worked out for each of these model category/category of fibrant object structures (and all these notions will be suitably equivalent).
An explicit model of such path space objects for $L_\infty$-algebras is discussed by Dolgushev in section 5 of his article arXiv:0703113.
See on the nLab at model structure for L-infinity algebra -- Homotopies and derived hom spaces.