Let $(A, {m_k}_{ok geq 1} )$ be an $A_{infty}$ algebra over a subject $ok$. Recall that that is the info of a $mathbb{Z}$-graded $ok$-vector area, together with a group of $k-$nary operations which fulfill “homotopy associativity” relations.

Let $overline{BA}:= oplus_{l geq 1} A[1]$ be the (decreased) bar complicated, the place $A[1]_j:= A_{j+1}$. Let $hat{m}_k: overline{BA} to A[1]$ be outlined by setting for $n geq ok$ $$hat{m}_k(x_1otimesdots otimes x_n)= sum (-1)^x_1 x_1otimesdotsotimes x_i otimes m_k(x_{i+1}otimesdotsotimes x_{i+ok}) otimes x_{i+ok+1} otimes dots otimes x_n)$$
and $hat{m}_k=0$ in any other case.

Let $hat{m}:= sum_1^{infty} hat{m}_k$. Then one can present the next:

Proposition: that $hat{m}$ is a co-derivation. In truth, there’s a bijection between co-derivations on $overline{BA}$ and $A_{infty}$ buildings on $A$.

Query: Suppose that my $A_{infty}$ algebra is $C_*(Omega X)$ for some topological area $X$. Then what’s $overline{BC_*(Omega X)}$ and what’s $hat{m}$? A guess can be that $overline{BC_*(Omega X)}$ is (decreased) chains on $X$, with the co-product given by the Alexander-Whitney map. What’s then $hat{m}$ on this context?