A-infinity space is originally taken in the sense of the homotopy in the usual topological category (Strom model structure if you want), not in the nicified ones, like simplicial sets or like infinity groupoids which model it only up to **weak** homotopy equivalence.

In email discussion with somebody I wanted to point to the $n$Lab entry *A-infinity space* only to notice that there is not much there. I have now spent a minute adding just a tiny little bit more…