There is a theorem saying that for any space $X$ which is connected and has the homotopy type of a CW complex, $X$ has the homotopy type of a looped space $\Omega Y$ if and only if $X$ has an $A_\infty$ structure. It is said that this result follows from the article Homotopy Associativity of H-spaces. I,II by James Stasheff, but I have not found an explicit proof in it. Could someone give me a guide on how to prove it?
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I don't think this has a short proof but i might be wrong. You might want to take a look at the book "The Geometry of Iterated Loop Spaces" by May. In this book he proves May's recognition principle which shows that a space is weakly equivalent to an $n$-fold loop space iff it is an $E_n$-algebra. The case for $n=1$ is the one you want to prove. I think the theorem is stated in section 1 and proven in section 13 of the book. – Frederik Jun 20 '21 at 17:25