Let $X$ be a topological space, and $X^X$ the category with one object ($X$) and morphism homeomorphisms of $X$.

(1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $X^X$?

(2) Does $H_*(X)$ also have an $A_\infty$-module structure? Is it the same as that of $H^*(X)$?

Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$.

(1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$?

(2) Does $H_*(X)$ also have an $A_\infty$-module structure? Is it the same as that of $H^*(X)$?

Added later: Jeff Giansiracusa gave a nice answer to (1). But his answer uses the ring structure in cohomology, leaving (2) open: Is there an $A_\infty$ structure on homology as well?