On a generalized homotopy transfer theorem

Question

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of this theorem, which I don't know if already exists (or even if it's true).

My starting operad $\mathcal{P}$ is a dg-operad (note that, in the version of the HTT I mentioned, the starting operad does not have a differential), not necessarily Koszul (if this makes sense). Let $A$ be a dg-$\mathcal{P}$-algebra, and $i: H \to A$, $p: A \to H$, $h: H \to H$ a homotopy retract of $A$. Is it true that $H$ has a $\Omega B \mathcal{P}$-algebra structure such that $i: H \to A$ is a quasi-isomorphism (or leads to a zig-zag of quasi-isomorphisms)?

Note: Here $\Omega B \mathcal{P}$ denotes the bar-cobar construction of the dg-operad $\mathcal{P}$.

My thoughts on this are that it is true, and that the formulas given in the book of Loday and Vallette for the construction of the homotopy algebra on $H$ could be extended in this context. However, I would like to know if this is already proven (so I can avoid writing a proof with a horrendous notation), or if this is known to be false (in which case I would love to see a counterexemple!).

Thanks in advanced :)

Answer 1

Suppose we are working over a field of characteristic 0. Then the category of dg-operads admits a projective model structure where are all objects are fibrant. Recall that a $\Omega B P$-algebra structure on $C$ is the same as a map of operads $\Omega B P \rightarrow \mathrm{End}(C)$. If $C \simeq D$ as chain complexes, it is relatively easy to show there is a zigzag of equivalences of operads $\mathrm{End}(C) \simeq \dots \simeq \mathrm{End}(D)$, and a little less obvious to see this zigzag can be taken through endomorphism operads. This relies on the fact that endomorphism operads are functorial with respect to retractions (not homotopy retractions), and so it suffices to connect the chain complexes by a zigzag of retractions which are quasiisomorphisms.

Thus, there is a sequence of arrows $\Omega B P \rightarrow \mathrm{End}(C) \simeq \dots \simeq \mathrm{End}(D)$. Because $\Omega B P$ is cofibrant, there is a map, canonical up to homotopy, $\Omega B P \rightarrow \mathrm{End}(D)$ and this can be compared to the original map $\Omega B P \rightarrow \mathrm{End}(C)$.

Answer 2

The Homotopy Transfer Theorem (HTT) applies in great generality. Morally, any algebraic structure encoded by a cofibrant operad can be transferred along weak equivalences between bifibrant objects. Connor's answer is an instance of this. For a general account see:

[Fresse: Props in model categories and homotopy invariance of structures, Part II]

Loday-Vallette's formulas, which have precedents in the literature, are also available for non-Koszul operads (in the dg-setting). I recommend you to have a look at the survey:

[Vallette: Algebra+Homotopy=Operad, Section 4]

As you can extract from there, there is a one-to-one correspondence between $\Omega\mathcal{C}$-algebra structures on $A$ (i.e. maps of operads $\Omega\mathcal{C}\to End_{A}$) and maps of cooperads $\mathcal{C}\to B(End_{A})$ for any (conilpotent) cooperad $\mathcal{C}$. Then, the HTT with explicit formulae follows from the following observation: the homotopy retract data $(i,p,h)$ yield an explicit map of cooperads $B(End_A)\to B(End_H)$; see Lemma 23 in loc.cit.. Then, $H$ is an $\Omega\mathcal{C}$-algebra via the map of cooperads $$ \mathcal{C}\to B(End_{A}) \to B(End_{H}). $$ So, this applies to $\mathcal{C}=\mathcal{P}^{¡}$ if $\mathcal{P}$ is Koszul, but also to $\mathcal{C}=B\mathcal{P}$.

Let me also mention that there are tons of papers dealing with generalizations of this. In particular, you should have a look at

[Berglund, Homological perturbation theory for algebras over operads]