When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism.

**Recall/Example ($A_\infty$-algebras):**

An **$A_\infty$-morphism** between two $A_\infty$-algebras $(A,\mathfrak{m})$ and $(A', \mathfrak{m}')$ (here $\mathfrak{m}$ and $\mathfrak{m}'$ are the structure maps) is a collection $\{f_k\}_{k\geq1}:(A,\mathfrak{m}) \rightarrow (A',\mathfrak{m}') $ of degree zero (degree preserving) multilinear maps
\begin{equation*}
f_k: A^{\otimes k}\rightarrow A', \hspace{1cm}k\geq 1
\end{equation*}
that satisfy the following relation for $n\geq1$:
\begin{equation*}
\sum_{k+l=n+1}\sum^k_{i=0} (-1)^{a_1+\dots+a_n}f_k(a_1, \dots, a_i, m_l(a_{i+1}, \dots, a_{i+l}), a_{i+l+1}, \dots, a_n).
\end{equation*}
\begin{equation*}
=\sum_{\substack{1\leq k_1\leq \dots \leq k_j \\ k_1+\cdots+k_j= n}} m'_j(f_{k_1}(a_1, \dots, a_{k_1}), f_{k_2}(a_{k_1+1}, \dots, a_{k_1+k_2}), \dots, f_{k_j}(a_{k_{j-1}+1}, \dots, a_n))
\end{equation*}

Furthermore, we call such morphisms **$A_\infty$-quasi-isomorphisms** if $f_1$ induces isomorphism in cohomology.

**Q1:** Why do we normally omit higher arity maps when talking about quasi-isomorphisms?

**Q2:** Would it be possible to have a weak equivalence that only appears in higher arity maps?

**Q3:** In case we only care about $f_1$, wouldn't that imply an equivalence at the level of homotopy categories between, for example, **Ho**(DGLA) and **Ho**(L$_\infty$), as all the higher arity maps between $A$ and $B$ in L$_\infty$ with the same $f_1$ give isomorphism in **Ho**(L$_\infty$)?