A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n = 2,3,\dots$ that are inductively described (the definition of $c_n$ uses $c_{n-1}$ as input; the map $c_2$ is a co-$H$ structure). The suspension of a based space $X$ has the structure of a co-$A_\infty$ space.
Assume $Y$ is $2$-connected and has the homotopy type of a finite complex. Then Schwaenzl, Vogt and I showed that a co-$A_\infty$ space $Y$ desuspends to a space $X$ in the sense that there's a weak equivalence $\Sigma X \simeq Y$.
However we didn't try to check that the given weak equivalence is compatible in the co-$A_\infty$ sense. Part of the problem is that a morphism $f: Y \to Z$ of co-$A_\infty$ spaces should amount a co-$A_\infty$-structure on its mapping cylinder restricting to the given ones on $X \times 1$ and $Y$. However, this doesn't form a category: it's an $\infty$-category.
Now to my questions:
Question 1: is there a documented proof somewhere that the functor which assigns to a based space $X$ its suspension (considered as an co-$A_\infty$ space) induces an equivalence between the homotopy category of $1$-connected spaces and $2$-connected co-$A_\infty$ spaces?
Presumably, such a proof should be Hilton-Eckmann dual to one of the main results in the Book of Boardman and Vogt.
Question 2: Do function spaces coincide up to weak equivalence under this functor? That is, is the map $$\hom_{\text{Top}_*}(X,X') \to \hom_{\text{co-}A_\infty}(\Sigma X,\Sigma X')$$ A weak equivalence under suitable hypotheses on $X$ and $X'$?
By $\hom$ in each case, I mean topologized mapping spaces.
How would one go about proving a result like this?
