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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.

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    $\begingroup$ It seems that what breaks is not the transference of $A_\infty$ structures, but earlier: quasi-isomorphisms need not have completions to homotopy equivalences when you do not have the axiom of choice at your disposal. Certainly the formulae work whenever you do have a homotopy equivalence $i,p,h$ connecting $A$ with $H(A)$. $\endgroup$
    – Theo Johnson-Freyd
    May 25, 2011 at 14:39
  • $\begingroup$ @Theo. Agreed. The point is that you do not always have $i$, $p$, $h$ connecting $A$ and $H(A)$. Bu still, there should be a way to transfer (actually this is exactly what the paper of Sagave mentioned by Fernando is about. $\endgroup$
    – DamienC
    May 26, 2011 at 23:01
  • $\begingroup$ @DamienC Please let me know if you figure out this sheaf thing! I was trying to get to the same issue at mathoverflow.net/questions/46373/… but I am still confused about it. You once told me that maybe it should involve a simplicial category of cofibrant dg algebras. $\endgroup$
    – Oren Ben-Bassat
    Sep 16, 2011 at 17:28
  • $\begingroup$ I meant to say dg coalgebras. $\endgroup$
    – Oren Ben-Bassat
    Sep 16, 2011 at 17:44
  • $\begingroup$ Dear Oren, yes I think I figured it out. I'll post an answer to me own question about it as soon as possible. Best, Damien $\endgroup$
    – DamienC
    Sep 16, 2011 at 18:01

1 Answer 1

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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...

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    $\begingroup$ @Fernando, Ok over a field one can always choose representatives of homology classes. But, if you are given a homotopy retract, then the transfer theorem works in any case, no matter the ground field being a field. $\endgroup$
    – Bruno V.
    May 26, 2011 at 22:55
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    $\begingroup$ @Bruno: you are right. But the point is precisely that you don't always have a homotopy retract in case you work with a ring k (or with sheaves). $\endgroup$
    – DamienC
    May 26, 2011 at 22:58
  • $\begingroup$ Hi Bruno! Nice to see you here :-) $\endgroup$
    – Fernando Muro
    May 27, 2011 at 10:03

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