Homotopy transfer in the opposite direction

Question

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy between the identity and $Y\rightarrow X\rightarrow Y$, and a couple of less relevant vanishing formulas hold. Moreover, let $\mathcal O$ be a Koszul DG-operad and $\mathcal O_\infty=\Omega\mathcal O^{¡}$ its resolution. It is well known how to transfer an $\mathcal O_\infty$ algebra structure on the big complex $Y$ to the small complex $X$, even with explicit formulas in terms of trees due to Kontsevich and Soibelman. Do you know of any written account where this is done in the opposite direction, that is, transferring an $\mathcal O_\infty$ algebra structure on the small complex $X$ to the big complex $Y$?

I'd like to remark that I'm interested in Kontsevich-Soibelman-like formulas. Otherwise, the existence of such transfers follows easily from standard model category arguments. Please, assume that we're working over a field of characteristic zero, if this simplifies things.

Answer 1

Let us denote by $p\colon Y\to X$ and $i\colon X\to Y$ the maps of your SDR. Since $pi=\mathop{\mathrm{id}}\nolimits_X$, the map $i$ is injective, and is an isomorphism with its image. The map $\pi=i\circ p$ is a projector (that is, $\pi^2=\pi$); this projector implements a splitting $Y=\ker(\pi)\oplus\mathop{\mathrm{Im}}(\pi)$. Of course, the image of $\pi=i\circ p$ coincides with the image of $i$ since $\pi\circ i=i\circ\pi\circ i=i$.

Recall that for a Koszul operad $\mathcal{O}$, the operad $\mathcal{O}_\infty$ is the cobar construction of the Koszul dual cooperad $\mathcal{O}^{\ ¡}$, and the structure of a $\mathcal{O}_\infty$-algebra on a chain complex $V$ is equivalent to a Maurer-Cartan element in the dg Lie algebra $$\mathfrak{g}_{\mathcal{O},V}:=\prod_{n\ge 1}\mathop{\mathrm{Hom}}\nolimits_{\mathbb{S}}(\overline{\mathcal{O}^{\ ¡}}(n),\mathop{\mathrm{End}}\nolimits_V(n)).$$ Let $\alpha\in\mathfrak{g}_{\mathcal{O},X}$ be the Maurer-Cartan element encoding the given $\mathcal{O}_\infty$-algebra structure on $X$. Consider the map of dg Lie algebras $$ i_*\colon \mathfrak{g}_{\mathcal{O},X}\to \mathfrak{g}_{\mathcal{O},i(X)}=\mathfrak{g}_{\mathcal{O},\pi(Y)}\subset \mathfrak{g}_{\mathcal{O},Y}. $$ This map sends a Maurer-Cartan element $\alpha$ to some Maurer-Cartan element in the algebra $\mathfrak{g}_{\mathcal{O},Y}$, which corresponds to an $\mathcal{O}_\infty$-algebra structure on $Y$. (This is almost verbatim the construction used when dealing with minimal models for $\mathcal{O}_\infty$-algebras. The difference between the two is that if $X$ is not minimal, then $\ker(\pi)$ might be just a part of the space on which the structure would vanish.)