For an $A\_\infty$ algebra (say I do everything over $\mathbb{Q}$) modeled by dg-algebras I can understand the definition of formality (that its homology groups is equivalent in a $A\_\infty$ way to the original algebra). I imagine that this definition also extends to the Lurie/spectra setting? As in take homotopy groups and ask whether the algebra I get is isomorphic to the original one as an $E\_1$-algebra. So with all this in mind, let me ask a very basic question. What is an example of a non-formal $A\_\infty$ algebra over $\mathbb{Q}$/a discrete ring?



What is an example of a non-formal $A_\infty$ algebra over a discrete ring?
Anyone with a non-trivial Massey product, e.g. the rational cochain algebra of the complement of the Borromean rings in $S^3$. Presented in this way, it is strictly associative. If you take a minimal model, the ternary operation $m\_3$ can't be trivial because it represents triple Massey products.


