Let $\mathcal{P}_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Let $A=A^{1}\oplus A^{2}$ be a graded vector space concentrated in degree 1 and 2. Let $m_{n}\: : \:{A^{1}}^{\otimes n}\to A^{2}$ be a family of linear maps (of degree 2-n) . These map corresponds to a coderivation $$\delta^{A}\: : \: \left( T^{c}\left(A[1] \right)\right)^{0}\to\left( T^{c}\left(A[1] \right)\right)^{1}.$$ Let $B$ be a ordinary positively graded $P_{\infty}$ algebra. Assume that there are graded coalgebra maps $F^{0}, F^{1}$ such that $$F^{0}\oplus F^{1}\: : \: \left( T^{c}\left(A[1] \right)\right)^{0}\oplus\left( T^{c}\left(A[1] \right)\right)^{1}\to \left( T^{c}\left(B[1] \right)\right)^{0}\oplus\left( T^{c}\left(B[1] \right)\right)^{1}$$ such that $\delta^{B}F^{0}=F^{1}\delta^{A}$ and graded coalgebra maps $G^{0}, G^{1}$ such that $$G^{0}\oplus G^{1}\: : \: \left( T^{c}\left(B[1] \right)\right)^{0}\oplus\left( T^{c}\left(B[1] \right)\right)^{1}\to \left( T^{c}\left(A[1] \right)\right)^{0}\oplus\left( T^{c}\left(A[1] \right)\right)^{1}$$ such that $\delta^{A}G^{0}=G^{1}\delta^{B}$
My questions:
1) Is it possible to extend $(\delta^{A},A)$ to a $P_{\infty}$-algebra $(\delta',A')$ (by eventually adding elements of higher degree) and $F^{0}\oplus F^{1}$ to a $P_{\infty}$-map?
2) Is it possible to extend $(\delta^{A},A)$ to a $P_{\infty}$-algebra $(\delta'',A'')$ and $G^{0}\oplus G^{1}$ to a $P_{\infty}$-map?
I think that the answer is yes by obstruction theory, but I would like to know more details.
