On the page 4 of [these notes](https://pdfs.semanticscholar.org/4674/0dee0b88cafbdcc5160f9c3c60be373eb2df.pdf) it is stated that an $A\_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A\_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to a non-unique $A\_{\infty}$-isomorphism. My question, is the morphism between the underlying vector spaces induced by this non-unique isomorphism unique?



Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical?
A minimal model of an $A\_\infty$-algebra $A$ is an $A\_\infty$-algebra $B$ with trivial differential together with an $A\_\infty$-quasi-isomorphism $B\to A$. This gives an isomorphism $H^\*A=H^\*B$, and we have $H^\*B=B$ since $B$ has trivial differential. Together these show that the underlying vector space of $B$ is indeed *canonically* isomorphic to the homology of $A$.


(Instead of $B$, you wrote $HA$ for the minimal model. That's of course correct in a sense, as the underlying vector space of the minimal model is indeed *canonically* the homology of $A$, but it is sure to cause confusion when explaining the above.)


