# Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?

If (say) $$L$$ is an $$L_\infty$$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $$L_\infty$$-algebra $$L'$$ which is a differential graded Lie algebra: all ternary and higher brackets of $$L'$$ vanish. This is sometimes called rectification.

(I am a physicist, so I assume $$L$$ is over a field of characteristic $$0$$; quasi-isomorphisms are then morphisms whose linear parts induce isomorphisms on cohomology. I wouldn't mind hearing about $$L$$ over formal power series rings, however.)

There exist lots of operad-powered general proofs, such as in this paper by Berger and Moerdijk. Unfortunately, those are not accessible to me, so I am looking for operad-free explicit constructions/algorithms producing $$L'$$ and the associated quasi-isomorphism. I am also interested in the case where $$L$$ is endowed with a cyclic inner product, should that make a difference.

• The "bar-cobar resolution" does precisely what you want, do you know of it? Feb 16, 2021 at 7:11
• A reference (though arguably as operadic as they come) is Theorem 11.3.6 of irma.math.unistra.fr/~loday/PAPERS/LodayVallette.pdf - unwrapping the definitions, the canonical dgla resolution of a $L_\infty$ algebra $A$ is the free (graded) Lie algebra on $(\bigoplus_{k>0} \Lambda^k A)[k-1]$, where the differential on generators is given by the sum of the Chevalley-Eilenberg differential (which lands in generators) and the reduced coproduct (which lands in brackets of generators). The map down to $A$ is induced by the projection to the summand labeled by $k = 1$. Feb 16, 2021 at 7:37
• @AlexArvanitakis in turn, I'd love to hear about the interest on Lie and L-infinity algebras over formal power series rings in physics. They need not have minimal models in the classical sense and the whole theory is a little bit more complicated, but a student of mine worked on this for his thesis and learning about potential new applications would be nice :) Feb 16, 2021 at 11:57
• @DanPetersen: I've heard of "bar construction" in this context, given in an off hand comment arxiv.org/abs/0705.3719... Feb 16, 2021 at 13:15
• ... which I guess alludes to the proof in @BertramArnold 's link. Also thanks Bertram, I can parse that definition. Feb 16, 2021 at 13:19