# Obstruction theory on $A_{\infty}, C_{\infty}$-algebras

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.

• Ciao Cepu. Notice that $A$ is already a $P_\infty$-algebra if you set $m_n=0$ wheneverany of the arguments has degree $2$ (since in that case you would land in degree $3$ or more). I don't know if you can extend the maps, but if you work with just $A$ it might be easier than extending the structure to have elements of higher degrees. – Daniel Robert-Nicoud Oct 13 '17 at 11:03