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There are two ways to define $A_\infty$ structures. The elementary one seems rather intuitive when I think of higher homotopies. I.e. an $A_\infty$ structure on a $\mathbb{Z}$-graded vector $A$ space is a sequence of maps $(m_n)_{n\in\mathbb{N}}$, where $m_n:A^{\otimes n} \to A$ is of degree $n-2$, satisfying the $A_\infty$-relations: $$\sum_{r+s+t=n-1}(-1)^{r+st}m_n\circ(id^{\otimes r}\otimes m_s\otimes id^{\otimes s})=0.$$

This definition is then usually "restated in a more efficient way". For example, in Keller's introductory paper:

Let $(\bar{T}A,\Delta)$ denote the reduced tensor coalgebra on $A$, then an $A_\infty$-stuctures is a degree $1$ coderivation that squares to zero.

This re-definition is very mysterious to me. What was the original motivation? How would Stasheff(?) have come up with it?

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    $\begingroup$ This actually works for other operads. You can restate the homotopy condition by using a (co)derivation. I had the same doubt some years ago, however I recall I could understand more or less the motivation when I saw a paper regarding minimality of operads which was a condition for writing the operad by such (co)derivation. I will see if I can find this paper again and I will post it here. $\endgroup$
    – user40276
    Dec 1, 2016 at 15:05
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    $\begingroup$ I think this was the paper arxiv.org/abs/hep-th/9411208 . $\endgroup$
    – user40276
    Dec 1, 2016 at 15:28
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    $\begingroup$ This should be an instance of Koszul duality (at least that's what the cool people say). I don't know how to make it precise, and if anyone had a good reference I'd love to read it. $\endgroup$
    – Denis Nardin
    Dec 1, 2016 at 15:45
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    $\begingroup$ Now I've found it. 10.1.17 in www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf states the condition explicitly for this correspondence using coderivations. More generally, 10.1.22 (Rosetta Stone) gives equivalent formulations for algebras up to homotopy. $\endgroup$
    – user40276
    Dec 1, 2016 at 15:55
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    $\begingroup$ @jds I'm not sure if the associative version or if the Lie version came first. However motivating one of these versions is enough since the latter is the infinitesimal version of the former. In any case, in group cohomology and Lie algebra cohomology the equation $d^2 = 0$ plus the Leibniz rule is equivalent to a representation and this was known long time ago. Now upgrading to graded vector spaces, the condition $d^2 = 0$ turns into the up to homotopy version of these equations. So I think in the end the motivation comes from the Chevalley-Eilenberg complex. $\endgroup$
    – user40276
    Dec 1, 2016 at 20:39

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