Definition of $A_\infty$-module

Question

Let $A$ be a $A_\infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_\infty$-module on $V$ the sequence of map $$ s_n : A^{n-1} \otimes V \to V, $$ where $s_0$ is the differential on $V$ and $s_n$ satisfy homotopy relations. Is it true that this definition is equivalent to the existance of $A_\infty$-morphism $$ s: A \to \operatorname{End}_k(V), $$ where $\operatorname{End}_k(V)$ is the dg algebra of $k$-linear endomorphisms of the graded module $V$?

Answer 1

Note that, by adjunction, the family of maps $s_n : A^{\otimes (n-1)} \otimes V\longrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{\otimes (n-1)} \longrightarrow \operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA \longrightarrow \operatorname{End}(V)$, where $BA$ is the bar construction of the $A_\infty$-algebra $A$. As you observe, this is the same as giving an $A_\infty$-map from $A$ to $\operatorname{End}(V)$. See the notes of B. Keller here.