Higher homotopy algebraic structure on the homology of an operad

Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is quasi-isomorphic to $A$, and moreover, this structure on the homology is unique up to isomorphism. The $A_\infty$ algebra structure is describe explicitly, as usual, by the collection of higher multiplications $m_n: H(A)^{\otimes n} \to H(A)$ (which are closely related to Massey products).

An operad in the category of chain complexes can be thought of as a generalisation of a DGA - we now have a sequence of complexes $P(n)$, and the $\circ_i$ compositions are associative products between these. The homology $H(P(n))$ is again an operad in chain complexes (with zero differential), and the above fact for DGAs should generalise to say that $H(P)$ carries the additional structure of a strongly homotopy operad for which it is quasi-isomorphic to $P$. Has this structure been describes explicitly in terms of higher $n$-ary analogues of the $\circ_i$ compositions? I would be very grateful if someone could point me towards an appropriate reference.

-
Seems to me like $H(P)$ should be an $A_\infty$-object in the category of symmetric sequences under the composition product - my guess is that you probably just replace the domain of $m_n$ with an n-fold composition product rather than a tensor. –  Tyler Lawson Jan 16 '11 at 3:13
Try looking at the work of Bruno Vallette? –  Kevin H. Lin Jan 16 '11 at 3:20
A short answer is yes, just as the higher A-infty maps are given as a twisting map from the free coalgebra on H(A) to H(A), the higher infinity operad maps are given as a twisting map from the free cooperad. I don't think I know the references well enough to give you a good long answer. There's a paper by Vallette and Merkulov which contains a lot of relevant background. Though perhaps the authors of most relevance are Chuang and Lazarev. I'm sorry that I can't be more precise. –  James Griffin Jan 17 '11 at 11:02
add comment

1 Answer

You are completely right Jeffrey. Since an operad is (strictly speaking) a generalization of the notion of associative algebra, there exists a homotopy transfer of structure through homotopy equivalences. This has been written, with explicit tree formulae, by Johan Granåker in "Strong homotopy properads" available at http://arxiv.org/abs/math/0611066 and published in IMRN. To get the statement on the level of operads, remove the two first letters in the word "properad" and consider only rooted trees instead of graphs in loc. cit. (More seriously, an associative algebra is an operad concentrated in arity 1. Properads model operations with several inputs and several outputs. So an operad is a properad concentrated in arity (many [inputs], 1 [outputs])).

In the paper with Merkulov, we use the homotopy transfer theorem for co(pr)operads to understand the minimal model of the properad encoding associative bialgebras (we apply it to the bar construction). And in a forthcoming paper with Gabriel Drummond-Cole, we use the same idea to make the minimal of the operad encoding Batalin-Vilkovisky algebras explicit.

-
add comment