To describe homotopy invariant algebraic structures on spaces, there are different approaches.


* The Stasheff / Boardman–Vogt / May approach, where operations and equations are replaced by spaces of operations, witnessing higher homotopy coherence.
* The Segal approach, where the structure maps that would be isomorphisms (in the strict version) are merely required to be weak equivalences.


In the introduction to [1], Badzioch writes: "Also, $A\_{\infty}$-spaces can be viewed as homotopy algebras over the algebraic theory $T\_{\mathrm{Mon}}$ such that the corresponding strict algebras describe monoids." He then points out that his main result specialized to the case of monoids yields an equivalence of homotopy theories $$h\mathrm{Alg}\_{\mathrm{Mon}} \simeq \mathrm{Alg}\_{\mathrm{Mon}}$$ between homotopy monoids à la Segal and (strict) topological monoids. Combined with the equivalence $$A\_{\infty}-\mathrm{Spaces} \simeq h\mathrm{Alg}\_{\mathrm{Mon}}$$ which he alluded to, this provides an alternate proof of the known equivalence $A\_{\infty}-\mathrm{Spaces} \simeq \mathrm{Alg}\_{\mathrm{Mon}}$ explained beautifully [here](https://mathoverflow.net/questions/51777/is-an-a-infinity-thing-the-same-the-same-as-strict-thing-viewed-through-a-homoto).



> 
> **Question.** Is there a good reference explaining the equivalence
>  $$A\_{\infty}-\mathrm{Spaces} \simeq h\mathrm{Alg}\_{\mathrm{Mon}}$$
>  between $A\_{\infty}$-spaces and homotopy monoids in spaces à la Segal? (Without going through strict topological monoids.)
> 
> 
> 


For this example, I can imagine an equivalence constructed more or less by hand. A functor in the Segal-to-Stasheff direction would involve some choices.


References to more general results of the form "homotopy algebras à la Stasheff $\simeq$ homotopy algebras à la Segal" would be welcome.


Here are some references I've looked at.


* Other papers by Badzioch as well as papers by Bergner [2][3] contain related material, but I couldn't find the answer to the specific question above.
* I believe that Balzin has worked on the distinction between the Stasheff approach and the Segal approach, more specifically some [advantages of the Segal approach](http://recherche.math.univ-angers.fr/spip.php?article29&site=56&session=15&presentation=153) in some situations.
* These [notes by Leinster](https://arxiv.org/abs/math/0002180) look at the problem from a (higher) categorical perspective. Section 3.4 suggests that the answer is not obvious.


[1] *Badzioch, Bernard*, [**Algebraic theories in homotopy theory**](http://dx.doi.org/10.2307/3062135), Ann. Math. (2) 155, No.3, 895-913 (2002). [ZBL1028.18001](https://zbmath.org/?q=an:1028.18001).


[2] *Bergner, Julia E.*, [**Simplicial monoids and Segal categories.**](http://www.arxiv.org/abs/math/0508416), Davydov, Alexei (ed.) et al., Categories in algebra, geometry and mathematical physics. Conference and workshop in honor of Ross Street’s 60th birthday, Sydney and Canberra, Australia, July 11--16/July 18--21, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3970-6/pbk). Contemporary Mathematics 431, 59-83 (2007). [ZBL1134.18006](https://zbmath.org/?q=an:1134.18006).


[3] *Bergner, Julia E.*, [**Rigidification of algebras over multi-sorted theories**](http://dx.doi.org/10.2140/agt.2006.6.1925), Algebr. Geom. Topol. 6, 1925-1955 (2006). [ZBL1125.18003](https://zbmath.org/?q=an:1125.18003).



Homotopy invariant structure: Stasheff versus Segal
As requested, my comments in the form of an answer:


At the end of *Categories and Cohomology Theories* Segal gives a fairly detailed sketch of how to compare these theories in the harder, $\mathbb{E}\_{\infty}$ case. The same sketch works in the $\mathbb{A}\_{\infty}$-case. Later, May-Thomason elaborated on Segal's remarks and gave an axiomatic treatment of infinite loop space machines.


Thomason, later still, then wrote down a comprehensive treatment of the *single* loop space results, which should in particular provide a published reference for what you have in mind.


That said, the functor going in one direction is easy to write down: associated to an $\mathbb{A}\_{\infty}$-space $X$ one has the simplicial space $[n] \mapsto \mathbb{A}\_{\infty}(n) \times X^{\times n}$, i.e. the bar construction. This is evidently a $\Delta$-space, or homotopy monoid, in the sense of Segal. From here one could construct an inverse a la Segal, May-Thomason, and Thomason, or else proceed in any number of ways...


