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$\_\infty$)?



Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?
Q1: The conventional theory of homotopy algebras is built on the premise that the lower-degree operations dominate over the higher-degree ones, in some sense. A discussion of this can be found in the introduction to my preprint "Weakly curved $\mathrm A\_\infty$-algebras over a topological local ring", <http://arxiv.org/abs/1202.2697>. This does not answer your question fully, but explains the underlying ideology to some extent.


Q2: The theory of curved DG-algebras/curved DG-coalgebras and the co/contraderived categories of CDG-modules/comodules/contramodules over them is built on the premise that the (co)multiplication dominates over the differential (and the differential dominates over the curvature). So the higher-degree operations dominate over the lower-degree ones in these "theories of the second kind". On CDG-coalgebras or DG-coalgebras, there is even a model structure with such weak equivalences (though a precise definition is more complicated and maybe does not accord to what you describe in the question). See my memoir "Two kinds of derived categories, ...", <http://arxiv.org/abs/0905.2621>.


Q3: It is indeed true that the homotopy category of DG Lie algebras is equivalent to the homotopy category of $\mathrm L\_\infty$-algebras, though perhaps for reasons more complicated than described in the question. Similarly, the homotopy category of associative DG-algebras is equivalent to the homotopy category of $\mathrm A\_\infty$-algebras.


Basically, with any $\mathrm A\_\infty$-algebra one can naturally associate a much bigger DG-algebra quasi-isomorphic to it; while for any DG-algebra one can, if one wishes, construct a (generally speaking) much smaller $\mathrm A\_\infty$-algebra quasi-isomorphic to it (in addition to the obvious option of viewing a DG-algebra as an $\mathrm A\_\infty$-algebra with vanishing higher operations).


But of course, one cannot just forget the higher operations of an $\mathrm A\_\infty$-algebra and obtain a quasi-isomorphic DG-algebra, firstly because the multiplication in an $\mathrm A\_\infty$-algebra need not be associative, and secondly, even if it is, the identity map cannot be extended to an $\mathrm A\_\infty$-morphism (in most cases).


