Question

Background

Suppose we are given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $\pi$ of $(g,Q)$. One can twist $(g,Q)$ with the Maurer-Cartan element $\pi$ and obtains a new $L_\infty$-algebra that we call $(g,Q_\pi)$. Furthermore, we can construct a Maurer-Cartan element $\pi'$ of $(g',Q')$ by the formula

$\pi' = \sum_{n=1}^\infty \frac{1}{n!} F_n(\pi, \ldots , \pi)$,

where $F_n$ is the $\bigwedge^n g \rightarrow g'$-part of $F$. I don't know whether there is a (better) term, so I call $\pi$ and $\pi'$ associated Maurer-Cartan elements

One can twist the morphism $F$ with the Maurer-Cartan elements $\pi$ and $\pi'$ and obtain an $L_\infty$-morphism $F_\pi$ from $(g,Q_\pi)$ to $(g',Q'_{\pi'})$. The references I found are Dolgushev: A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold (section 2.4) and Yekutieli: Continuous and Twisted L_infinity Morphisms (section 3).

Question

Given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$, an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$, a Maurer-Cartan elements $\pi$ of $(g,Q)$ and a Maurer-Cartan element $\omega$, $\omega\neq \pi'$, of $(g',Q')$. Can one construct an $L_\infty$-morphisms between $(g,Q)$ twisted with the Maurer-Cartan element $\pi$ and $(g',Q')$ twisted with the Maurer-Cartan element $\omega$, where the Maurer-Cartan elements are not "associated"? I.e. can one construct an $L_\infty$-morphism between $(g,Q_\pi)$ to $(g',Q'_\omega)$?

Answer 1

Here's an idea for simplifying the problem a bit. I don't know the technical details or whether it will work, but it wouldn't fit in the comment box, so I'll put it as an answer, even though it's incomplete.

First, answer this simplified question: If $\alpha$ is an MC-element of $(h,P)$, is there a morphism $(h,P_\alpha) \to (h,P)$ and a morphism $(h,P)\to (h,P_\alpha)$? (This corresponds to $\pi$ or $\omega$ being zero and $g=g'$, $Q=Q'$ in the original question.) Then compose the morphisms

$(g,Q_\pi) \to (g,Q) \to (g',Q') \to (g',Q'_\omega) $,

where the first arrow comes from $h=g$, $P=Q_\pi$, and $\alpha = -\pi$, the second is the given morphism, and the third comes from $h=g'$, $P = Q'$, and $\alpha = \omega$.