The definition of $L_\infty$-algebra is by now pretty standard. I gather that the sign conventions given in Lada–Markl's paper Strongly homotopy Lie algebras, Communications in Algebra 23 Issue 6 (1995) (arXiv:hep-th/9406095) are widely used, and I will keep to them here. I will not rehash the de...

From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC moduli space") correspond to deformations. (More accurately the MC moduli space is formulated as a...

We call a sequence of $L_\infty$-algebras (weak) maps $$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$ is exact if it is exact on the the underlying chain complexes level. Thought I don't know whether this is a good notion. A trivial case is exact sequence of Lie algebras. It is clear that gi...

I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter object is an $E_\infty$ algebra, and so, by the answer in Tensor products of $\infty$-algebras ov...

Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a strong deformation retraction $(i,p,k)$ of $W$ onto $V$, i.e. $p$ is a cochain map $W\to V$, $i$ is ...

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...