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



What is a homotopy between $L_\infty$-algebra morphisms
One standard answer\*, in which any reasonable (characteristic $0$ — I haven't thought about any other case) algebraic category can be given a simplicial structure, is the following.


Let $\mathbb Q[\Delta^k] = \mathbb Q[t\_0,\dots,t\_k,\partial t\_0,\dots,\partial t\_k] / \bigl\langle \sum t\_i = 1,\ \sum\partial t\_i = 0\bigr\rangle$ denote the differential graded commutative algebra (dgca) of polynomial forms on the standard $k$-simplex. Here $t\_i$ are in (co)homological degree $0$, and their derivatives $\partial t\_i$ are in degree $\pm 1$ depending on whether you prefer homological or cohomological conventions. It is straightforward to check that $\mathbb Q[\Delta^k]$ has (co)homology only in degree $0$, where it is $1$-dimensional. Moreover, there are natural face and degeneracy maps between different $\mathbb Q[\Delta^k]$, making $\mathbb Q[\Delta^\bullet]$ into a simplicial dgca.


Given two $L\_\infty$ algebras $V,W$ (or, really, objects of any reasonable category of "algebras"), one then defines the *space* of maps $V \to W$ to be the simplicial set
$$ \hom\_\bullet(V,W) = \hom(V,W[\Delta^\bullet]),$$
where $W[\Delta^\bullet] = W\otimes\_{\mathbb Q} \mathbb Q[\Delta^\bullet]$ is the $L\_\infty$ algebra $W$ base-changed to live over the $k$-simplex. It is reasonably straightforward to prove that this simplicial set satisfies the Kan horn-filling condition, at least when $V$ is "quasifree" — in particular, in your situation of "nonlinear $L\_\infty$-algebra homomorphisms", the Kan condition is always satisfied.


Before I spell this out, I'm going to change your notation. What you called $f\_k$ I will call $f^{(k)}$, since it plays the role of the "$k$th Taylor coefficient of $f$". That way, I can ask "what is a *homotopy* between two morphisms $f\_0,f\_1 : V \to W$ of $L\_\infty$-algebras?" 


The answer is the following data: (1) a (nonlinear) homomorphism $f\_t: V \to W$ that depends polynomially on a parameter $t$, with the correct evaluations $f\_t|\_{t=0} = f\_0$ and $f\_t|\_{t=1} = f\_1$; (2) maps $\phi^{(k)}\_t : V \to W[1]$ (or maybe I mean $[-1]$), also depending polynomially on the parameter $t$. These data must satisfy a certain ODE of the form:
$$ \frac{\mathrm d}{\mathrm d t} f\_t = \operatorname{ad}\_{f\_t}(\phi\_t) $$
Of course, this is really an infinite sequence of equations (which are equations to things that depend polynomially on $t$). The $k$th entry on the left hand side is $ \frac{\mathrm d}{\mathrm d t} f\_t^{(k)}$. On the right hand side, the $k$th entry is computed as follows (up to a sign which I don't feel like working out). Consider the equations saying that $f\_t$ is a homomorphism; one of these equations is an equation of things with $k$ inputs $x$. Sum over all ways to replace, in each summand in this equation, one of the occurrences of an $f$ by a $\phi$. Such a sum is what I mean by the right-hand side. In short-hand, what I mean is: there is (a sequence of) equations $M(f)$, such that $f$ is a homomorphism iff $M(f) = 0$. The right hand side is $\frac{\partial M}{\partial f} \cdot \phi$.


In good situations like yours, all the ODEs that occur when studying $\hom\_\bullet(V,W)$ are pretty well behaved. In particular, their integral forms are contraction mappings in the appropriate sense, so the initial and boundary value problems are pretty easy to analyze formally.




---


\*Here is an important (elementary) exercise to work out if you want to understand this "standard answer." Consider just the category of chain complexes. Then, for $k \geq 0$, $\pi\_k\bigl( \hom(V,W[\Delta^\bullet])$ is the space of chain maps $V \to W[\pm k]$ modulo chain homotopies, i.e. it is $\mathrm{H}\_k(\underline\hom(V,W))$, where $\underline\hom$ denotes the chain complex of all linear maps $f: V \to W$ with differential $f \mapsto [\partial,f] = \partial\_W\circ f -(-1)^{\deg f}f\circ \partial\_V$. (Whether the shift should be $[k]$ or $[-k]$, and whether I mean $\mathrm H\_{\pm k}$ or $\mathrm H^{\pm k}$ or ..., depend on your conventions, so I didn't work them out.)


