My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at some point.
First, the definition of an $E_\infty$-operad.
- For spaces, as considered by Peter May here, an $E_\infty$-operad is a symmetric operad $\mathcal{E} = \{\mathcal{E}(n)\}$ consists of spaces $\mathcal{E}(n)$ with a free action of the symmetric group $\Sigma_n$, and furthermore each $\mathcal{C}(n)$ is a contractible space.
- For chain complexes (or DGAs) over some ring $R$, an $E_\infty$-operad $\mathcal{E}$ is a symmetric operad where each $\mathcal{E}(n)$ is free over the group ring $R[\Sigma_n]$, and furthermore $\mathcal{E}(n)$ is quasi-isomorphic to $R$ (i.e, $\mathcal{E}$ is quasi-isomorphic to the operad $\mathcal{Com}$ of commutative $R$-algebras).
- In general, an $E_\infty$-operad is a $\Sigma$-free cofibrant operad, where some suitable model structure is chosen on the category of symmetric operads. In particular, the weak equivalences are given point-wise.
Now, let $\mathcal{E}$ be some fixed $E_\infty$-operad in chain complexes. Then, for two $\mathcal{E}$-algebras $A, B$, an $E_\infty$-morphism $f:A \to B$ is a chain map $f : A \to B$ satisfying the commutative diagram
$$\require{AMScd} \begin{CD} \mathcal{E}(n)\otimes A^{\otimes^n} @>>> A\\ @V{1\otimes f^{\otimes^n}}VV @VV{f}V \\ \mathcal{E}(n)\otimes B^{\otimes^n} @>>> B \end{CD}$$
This is the usual definition of a morphism for algebras over an operad. But in the case of, say, $A_\infty$-algebras, there is a well known notion of weak morphisms which satisfies many nice homtopical properties. In fact, this weak morphism is usually called an $A_\infty$-morphim in the literature (eg, section 3 of this article by B. Keller). If we follow the definition from the article Homotopy theory of homotopy algebras by Bruno Vallette (or the book Algebraic Operads by J.L. Loday and B. Vallete), this weak morphism should be called an $\infty\text{-}A_\infty$-morphism.
Question: Is there a suitable notion of $\infty$-morphisms for $\mathcal{E}$-algebras, where $\mathcal{E}$ is some fixed $E_\infty$-operad?
Motivation: The main motivation behind the question is a result of Gugenheim as proved here (and interpreted in the language of $A_\infty$-algebra here). It says that the Stokes' integration map $\int : \Omega^\bullet(M) \to C^\bullet(M)$ for a closed manifold $M$ can be lifted to an $\infty\text{-}A_\infty$-quasi-isomorphism, where $C^\bullet(M)$ is the complex of smooth cochains with $\mathbb{R}$-coefficients, and $\Omega^\bullet(M)$ is the deRham complex. Now, $C^\bullet(M)$ has a natural $E_\infty$-algebra structure via the surjection operad of McLure and Smith, and $\Omega^\bullet(M)$ is trivially an $E_\infty$-algebra. The $\int$-map induces an algebra map in the cohomology, but it fails to be an algebra map at the cochain level, which implies that $\int$ cannot be lifted to a strict $E_\infty$-morphism. But can we hope for a lift to a weak $E_\infty$-morphism? Something very similar was proved here for the Cartan-Serre map. Note that the authors are considering strict $E_\infty$-morphisms in the article, since the Cartan-Serre map is known to be an algebra morphism.
I would assume that an $\infty$-morphism $f:A \to B$ of $\mathcal{E}$-algebras should be given by a family of maps $$\{f_n : \mathcal{E}(n)\otimes A^{\otimes^n} \to B\}$$ satisfying some coherency conditions. Another possibility might be similar to Gugenheim and Munkholm's DASH morphisms: define an $\infty$-morphism $A_1 \to A_2$ between two $E_\infty$-algebras to be a strict $E_\infty$-morphism $BA_1 \to BA_2$, where $BA_i$ is the bar complex of $A_i$. The bar complex of an $E_\infty$-algebra is again an $E_\infty$-algebra as proved by B. Fresse in this article, but I am not sure if Fresse considers such maps as I am not too familiar with the content of the article. I have not looked into Jacob Lurie's work on Higher Algebra. If Lurie indeed considers such an $\infty$-morphism, I would be grateful if someone could point it out to me.
Assuming a suitable notion $\infty$-morphism exists, here are some properties that are desirable in my opinion.
For $\infty\text{-}A_\infty$-morphism, it is well-known that $\infty$-quasi-isomorphisms are invertible up to homtopy. Can we expect similar statement for $\infty\text{-}E_\infty$-morphisms?
A question about transfer of $E_\infty$-algebra structure was asked in this MO post. But I think there an "$E_\infty$-map" is just a strict $E_\infty$-morphism. Can we hope for a transfer theorem with this weaker notion of $E_\infty$-morphism?
Given an arbitray operad $\mathcal{P}$, one can consider the operad $\mathcal{P}_\infty = \Omega B \mathcal{P}$, where $B(\_)$ is the (operadic) bar construction and $\Omega(\_)$ is the cobar construction. Following Bruno Vallette's article, there is a notion of $\infty\text{-}\mathcal{P}_\infty$-morphisms in the category of $\mathcal{P}_\infty$-algebras. In particular, Vallette proves that the homotopy category of $\mathcal{P}$-algebras is equivalent to the category of $\mathcal{P}_\infty$-algebras, with $\infty\text{-}\mathcal{P}_\infty$-morphisms. Can we identify the category of $\mathcal{E}$-algebras (for a fixed $E_\infty$-operad $\mathcal{E}$) with $\infty\text{-}E_\infty$-morphisms as a homotopy category of some other category?
Any comments or references regarding this will be highly appreciated. Thanks!
