# homological algebra – References for Homotopy transfer problem

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?