1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Daniel Robert-Nicoud Oct 13 '17 at 11:03

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.