Homotopy of maps of $A_{\infty}$-algebras

Question

Let $f, g: A \to B$ be maps of $A_{\infty}$-algebras. What is the correct (explicit) notion of a homotopy between $f$ and $g$? This is given in the expository paper of Keller in terms of maps of the associated reduced tensor algebra, but no where in the literature does it specify conditions on the actual maps $f$ and $g$ to be homotopic.

I imagine we want to do the standard thing, which is take a map $h_n: A^{\otimes n} \to B$ of some degree (possibly $2-n$ since a map of $A_{\infty}$-algebras should have degree $1-n$), then $f$ is homotopic to $g$ if $(f-g)_n=h_n d_A + d_B h_n$ (note this is probably wrong since it ignores the presence of all the higher homotopies!).

Is there a reference where I can see an explicit definition?

Answer 1

This must be done in multiple places, but one is my paper with D.-M. Lu, Q.-S. Wu and J. J. Zhang, "$A_\infty$-algebras for ring theorists," Algebra Colloquium 11 (2004), 91-128. Quoting from that paper:

We say that an $A_\infty$-morphism $f: L\to M$ is nullhomotopic if there is a family of graded maps $$ h_n: A^{\otimes n-1}\otimes L\to M, \quad n \geq 1, $$ of degree $-n$ such that $$ f_n=\sum_{v=1}^n m_{1+n-v} \circ (id^{\otimes n-v}\otimes h_v) +\sum_{n=r+s+t, s\geq 1} (-1)^{r+st}h_{r+1+t} \circ (id^{\otimes r}\otimes m_s\otimes id^{\otimes t}). $$ Two $A_\infty$-morphisms $f,g: L\to M$ are said to be homotopic if $f-g$ is nullhomotopic.

As Keller suggests, it should be easy to translate his version to this one.