**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. 



homotopy transfer for sheaves of algebras
If $R$ is a commutative $k$-algebra, a quasi-coherent sheaf of dg-$k$-algebras on $\operatorname{Spec}R$ would be just a dg-$R$-algebra $A$. It's known that there need not be any $R$-linear A-infinity structure on $H(A)$ quasi-isomorphic to $A$ in the $A$-infinity sense. Therefore the classical transfer theorem (which only works over fields) does not sheafify. Nevertheless, Sagave defined the notion of *derived A-infinity algebra* in such a way that there is a derived A-infinity structure on $H(A)$ quasi-isomorphic to $A$ in an $R$-linear and derived A-infinity sense, see :


MR2608191 (2011c:16030)
Sagave, Steffen(N-OSLO)
DG-algebras and derived -algebras. (English summary)
J. Reine Angew. Math. 639 (2010), 73–105. 


This solves the case of quasi-coherent sheaves on affine schemes, so it could be a good starting point to answer your question...


