Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the other in Tamaroff's article, which agrees with Loday-Vallette's book Algebraic Operads). Of course the second definition is only restricted to dg algebras, while first one covers the more general concept of $A_{\infty}$-algebras.
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The two definitions are the same. The thesis of Lefèvre-Hasegawa does not require the differential to be zero, it requires the component $m_1$ of the differential to be equal to zero: minimality translates as not having linear part. This is of course the same definition as used in the article of Tamaroff and the book of Loday and Vallette.
answered Mar 31, 2021 at 16:01
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The way you can see the definitions are the same is by noting that one can pack up an $A_\infty$-algebra $(A,\mu_1,\mu_2,\ldots)$ into its bar construction $(BA,d)$ (or, dually, as in the article you refer to, pack up an $A_\infty$-coalgebra into a cobar construction, which is a free algebra).
Then the linear part of $d$ is precisely (a suspension) of $\mu_1$ and the condition that $d(V)\subseteq V^{\geqslant 2}$ is equivalent to the linear part of $d$ being zero. The $A_\infty$ interpretation is given by Lefèvre-Hasegawa, while the one using free (or cofree nilpotent co)algebras is in the book and article you refer to.
