homological algebra – References for Homotopy transfer problem

I am trying to read Algebra+homotopy=operad by Bruno Vallette.

Consider the following set up :

  • chain complexes $(A,d_A),(H,d_H)$,
  • a degree $1$ morphism of chain complexes $h:(A,d_A)rightarrow (A,d_A)$,
  • a pair of morphisms of chain complexes $p: (A,d_A)rightarrow (H,d_H)$ and $i:(H,d_H)rightarrow (A,d_A)$

such that $1_A-ip=d_Ah+hd_A$ (seen as a morphism of chain complexes).

Suppose further that there is an extra structure on the chain complex $(A,d_A)$, namely a morphism $v:A^{otimes 2}rightarrow A$ satisfying the associativity condition, that is $v(a,v(b,c))=v(v(a,b),c)$ for all $a,b,cin A$. It is not mentioned explicitly, but I am assuming this $v$ is a collection ${v_{n,m}:A_notimes A_mrightarrow A_{n+m}}$. Further, it is assumed that the differential $d_A$ is a derivation for the product.

Now, consider the composition $pcirc vcirc (i_notimes i_m):H_notimes H_mrightarrow H_{n+m}$.

One question is if these collection ${pcirc vcirc (i_notimes i_m)}_{n,min mathbb{Z}}$ combine to give an associative algebra structure on the chain complex $(H,d_H)$. This is what is called (at least what I think) the Homotopy transfer problem. It turns out that this does not give an associative algebra structure but this gives some identities that forms what is called an $A_infty$-algebra. This is how they introduce $A_infty$-algebra.

I am having trouble understanding the intuition behind those identities. This could be because I don’t have sufficient background.

Can some one suggest some references that starts from scratch on this Homotopy transfer problem?