Let $\varphi:A\to \mathrm{Ass}$ be an $A\_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$:


1. Rectify $X$ by taking the pushforward $\varphi\_!X$, which is now a topological monoid. Now take the classical bar construction $|N(\varphi\_!X)|$. This should be the notion of rectification taken from [[Berger/Moerdijk, 2003]](https://arxiv.org/abs/math/0206094v2).
2. Rectify $X$ by forming the 2-sided bar construction $B(\mathrm{Ass},A,X)$ which is again a topological monoid, and then take the classical bar construction $|N(B(\mathrm{Ass},A,X))|$. This is the description given in [[Basterra et. al., 2016]](https://arxiv.org/abs/1612.07791).
3. We see that the reduced suspension is a right $A$-module functor by the splitting
$$\Sigma A X \to \Sigma\left(\bigvee\_{k\ge 1}X^{\wedge k}\right)\to \Sigma X,$$
so we can directly form the 2-sided bar construction $B(\Sigma,A,X)$. For $A=\mathcal{C}\_1$, this should be the classical description from [[May, 1972]](https://www.math.uchicago.edu/%7Emay/BOOKS/geom_iter.pdf).


Is there any reference comparing these three descriptions? I would expect that (at least for well-pointed $X$), all descriptions agree up to (weak) equivalence?



Different ways to “deloop” a (topological) $A_\infty$-algebra
There are two old papers that address this topic in some detail: R. W. Thomason. Uniqueness of delooping machines. \url{https://projecteuclid.org/euclid.dmj/1077313403}
Z. Fiedorowicz. Classifying spaces of topological monoids and categories. \url{https://www.jstor.org/stable/2374307?seq=1#metadata\_info\_tab\_contents}. Thomason's paper compares 2 and my original version of 3 and others. I haven't looked up Berger and Moerdijk.


