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 definition of $L_\infty$-algebra, because I'm sure the people who can answer this question will know it well-enough, or could look it up. I emphasise that in my case, I am not assuming anything is finite-dimensional, so that I am unwilling to dualise from dg coalgebras to dg algebras in case there is something funny going on.
What I want to know is what is the correct set of conditions on a weak map of 2-term $L_\infty$-algebras. Such a thing is a dg-coalgebra map of the graded cocommutative cofree coalgebra equipped with the differential that is the sum of all the brackets of the $L_\infty$-algebra.
I have found several published papers that all use the definition of Lada–Markl for $L_\infty$-algebras, specialise to the case of 2-term $L_\infty$-algebras or slight variations, then give conflicting definitions for a morphism. None of them have commented on potential errors in the others, or if there are conventions that differ leading to different signs. The coherence conditions have various signs flipped in various ways, and it's not clear to me, a complete uninitiate for dealing with cofree coalgebras of infinite-dimensional vector spaces, how to even arrive at even one of these sets. All of the sources I have seen so far do not actually calculate these coherence conditions, merely state them as an definition, rather than deriving them from the definition I just gave (if there is a published source working this stuff out, I'd love to see it).
Note: The SE software is suggesting to me this question, except that uses an apparently different sign convention compared to Lada–Markl. However, that's exactly the sort of formula I'm after – just with the signs sorted (which is of course the whole difficulty in this game), and I don't trust myself to not miss something.
Edit I will point out that Lada and Markl do provide an example of an explicit formula for a class of weak maps, namely from a $L_m$-algebra (including $m=\infty$) to a dg-Lie algebra (i.e. an $L_\infty$-algebra with no brackets of arity higher than 2). This is Definition 5.2, and in Remark 5.3 they give their definition of weak map, merely as a map of dg coalgebras between the free coalgebras generated from the $L_\infty$-algebras, with the differential coming from extending the collection of brackets as a derivation. The Remark then claims that the collection of formulas in Definition 5.2 defines a weak map.
What I would like is to know if this is indeed consistent. No derivation is given, and maybe it's obvious. However, the signs are the tricky part, and I have no idea how the particular combination of signs in Definition 5.2 are arrived at.
Best Answer
In the article Classification of 2-term $L_\infty$-algebras (arXiv:2109.10202), Kevin van Helden gives the definition of a morphism of 2-term $L_\infty$-algebras (Definition 2.3), and he was kind enough to share with me some private calculations that go through and checks the definition agrees with the one given in Lada and Markl's Remark 5.3.