**homotopy transfer for algebras**

Let $A$ be a differential graded (dg) $k$-algebra, and $H(A)$ its cohomology. $H(A)$ is naturally equipped with the structure of a graded algebra. In general we don't have that $H(A)$ and $A$ are weakly equivalent (i.e. quasi-isomorphic).

Nevertheless, it is well-known that there exists an $A_\infty$-algebra structure $(m_k)_{k\geq1}$ on $H(A)$ with the following properties:

- $H(A)$ equipped with this $A_\infty$-structure is weakly equivalent to $A$.
- the first structure map $m_1$ (i.e. the differential) vanishes.
- the second structure map $m_2$ coincide with the natural product on $H(A)$.

This structure is essentially unique: it is unique up to a unique $A_\infty$-isomorphism.

Moreover, there are explicit formulas for for the $A_\infty$-structure and the weak equivalence, in terms of planar trees. The main point is that the formula involves the choice of quasi-isomorphisms $i:H(A)\to A$ and $p:A\to H(A)$, together with an homotopy $h$ between $i\circ p$ and $id_A$.

**homotopy transfer for sheaves of dg algebras?**

I would be interested to know how to adapt this for sheaves. Namely, if now $A$ is a sheaf of dg algebras and $H(A)$ its cohomology sheaf. First of all one has to assume that $A$ is formal (i.e. quasi-isomorphic to $H(A)$) as a sheaf of $k$-modules. But even in this situation the existence of $i$, $p$ and $h$ is not guarantied.

**the question**

How does homotopy transfer works for sheaves of algebras?

**possible (incomplete) answer**

One can do things locally and then try to glue, the gluing condition will probably be satisfied only in a weak sens... and then my question might boil down to

what is the right definition of a (homotopy) sheaf of $A_\infty$-algeras?

A way to answer this is to use model categories. I was planning to proceed in the following way (very shortly): $A_\infty$-algebras are fibrant objects in the model category of dg coalgebras, then we have a Reedy model structure on presheaves of dg coalgebras, and sheaves of $A_\infty$-algebras can be defined as fibrant objects in this model category.

At this moment I am stuck. I am sure I am not far from the answer, but somehow I can't see the point.