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edited May 18, 2015 at 14:57

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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=B\mathcal O^{¡}$$\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.

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 $\mathcal O_\infty=B\mathcal O^{¡}$ 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 $Y$?

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 $Y$?

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asked May 18, 2015 at 14:29

Fernando Muro

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Homotopy transfer in the opposite direction

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 $\mathcal O_\infty=B\mathcal O^{¡}$ 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 $Y$?