Consider based maps $f : X \to Z$ and $g : Y \to Z$, which induces the following map at the based loop space level : $$\theta := \mu_Z \circ \big(\Omega f \times \Omega g) : \Omega(X\times Y) = \Omega X \times \Omega Y \to \Omega Z \times \Omega Z \to \Omega Z,$$
where $\mu_Z$ is the standard choice of loop concatenation. Suppose we are given that $\theta$ is an $A_2$ map in the sense that there is a homotopy $$\mu_Z \circ (\theta \times \theta) \simeq \theta \circ \mu_{X\times Y},$$
where $\mu_{X\times Y}$ is the standard concatenation in $\Omega(X\times Y)$.
Question: Can we claim that $\theta$ is an $A_\infty$-map? In particular, assuming suitable connectivity of $X,Y,Z$, can we deloop $\theta$ to get a map $X\times Y \to Z$?
I understand it is probably too much to ask for! But here is my approach. I will consider the interval operad $\mathcal{E}_1$ as the $A_\infty$-operad acting on the loop spaces. From this, I am able to prove that the following diagrams are homotopy commutative for each $n$ :
$$\require{AMScd}
\begin{CD}
\mathcal{E}_1(n) \times \big(\Omega(X\times Y)\big)^n @>{}>> \Omega(X\times Y)\\
@V{1\times \theta^n}VV @VV{\theta}V\\
\mathcal{E}_1(n) \times \big(\Omega Z\big)^n @>>> \Omega Z
\end{CD}$$
This is weaker than the notion of $\mathcal{E}_1$-map as in The Geometry of Iterated Loop Spaces by J. P. May, and so I am unable to apply the recognition principle directly. On the other hand, I have come across articles in the algebraic context (e.g., this one), where the author considers the above as the definition of an $E_\infty$-coalgebra map. Also, at the very end of the same monograph by May, the author notes that the theory can possibly be weaken to include homotopy morphisms. In the article Strong homotopy algebras over monads by T. Lada, the author introduced a notion of strong homotopy morphism, which asks for higher homotopy relations, and proved the recognition principle. But the $\theta$ map is not a strong homotopy morphism (in the sense of T. Lada) either. Can we still expect a recognition principle without higher homotopy relations?
Any comment or reference regarding this will be highly appreciated.
Best Answer
No, take $Z$ a connected space for which $\Omega Z$ is homotopy commutative but $Z$ has no $A_\infty$ multiplication, e.g. $Z$ could be an $H$-space which doesn't have the homotopy type of a loop space. If $f=g=\mathrm{Id}_Z$, then the multiplication $\Omega Z \times \Omega Z \rightarrow \Omega Z$ is an $A_2$-map by the hypothesis of homotopy commutativity and and Eckmann-Hilton argument. If it were an $A_\infty$ map, we could apply the bar construction to achieve a map $B(\Omega Z \times \Omega Z)=B\Omega Z \times B\Omega Z \rightarrow B \Omega Z$ which is an $A_\infty$ multiplication. Since $Z$ is connected $B\Omega Z \simeq Z$, so we have produced an $A_\infty$ multiplication on $Z$ which contradicts our assumption.