Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping cylinder $$ M_f = Y \cup_{f\times 0} X\wedge (I_+) $$ which extends the given structures on $X$ and $Y$. These do not form a category because composition is only defined up to "contractible choice." However, they do form a weak Kan complex (or $\infty$-category); the $k$-simplices are formed by taking iterated mapping cylinders of a $k$-fold composition and then taking the $A_\infty$-structures on that which restrict to $A_\infty$-structures on all faces.

So we obtain a space of $A_\infty$-maps from $X$ to $Y$.

(Note: an $A_\infty$-map is not assumed to strictly commute with the operad actions on $X$ and $Y$ (when it does, it's called an $A_\infty$-homomorphism.)

We can define similar notions of "map" and "homomorphism" for $A_\infty$-ring spectra.

On the other hand, EKMM defined a category of structured associative ring spectra which is enriched over spectra.

** Question: ** If $R$ and $S$ are EKMM type associative ring spectra which are both fibrant and cofibrant, how does the EKMM version $\hom(R,S)$ relate to the Boardman-Vogt type notions?

More precisely, is it weak equivalent to the spectrum of $A_\infty$-homomorphisms in the Boardman-Vogt sense, or to the spectrum of $A_\infty$-maps in the Boardman-Vogt sense?