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](https://www.math.uchicago.edu/%7Emay/BOOKS/geom_iter.pdf) 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](https://arxiv.org/abs/2004.06652)), 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.



Delooping a weak $E_1$ map by bar construction
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.


