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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.

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 in some situations.

  • These notes by Leinster 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, Ann. Math. (2) 155, No.3, 895-913 (2002). ZBL1028.18001.

[2] Bergner, Julia E., Simplicial monoids and Segal categories., 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.

[3] Bergner, Julia E., Rigidification of algebras over multi-sorted theories, Algebr. Geom. Topol. 6, 1925-1955 (2006). ZBL1125.18003.

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    $\begingroup$ I assume you'd like an equivalence of homotopy theories, not literal categories, right? Segal himself gives a fairly detailed sketch of an equivalence in the harder, $E_{\infty}$ case at the end of his categories and coh. theories paper. In this case, one direction is easy: Given an $\mathbb{A}_{\infty}$-space $X$ form the bar construction with the $\mathbb{A}_{\infty}$-operad, $[n] \mapsto \mathbb{A}_{\infty}(n) \times X^{\times n}$. That is a $\Delta$-space or 'homotopy monoid' in the sense of Segal. You can either check directly that this is essentially surjective and $\endgroup$ – Dylan Wilson Mar 23 '18 at 19:34
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    $\begingroup$ (homotopically) fully faithful, or else make some choices and build an inverse. Another reference for the commutative story is May-Thomason... and I think Thomason has some separate paper about the monoid ("uniqueness of delooping machines") which probably contains this result? $\endgroup$ – Dylan Wilson Mar 23 '18 at 19:36
  • $\begingroup$ Thanks Dylan! Segal's Appendix B does look like what I'm looking for. May–Thomason and Thomason are also very helpful. Please make your comments into an official answer. $\endgroup$ – Martin Frankland Mar 23 '18 at 19:49
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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...

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