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While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, whith the other data suitably defined and such that its algebras are precisely the topological monoids).

enter image description here

More precisely, it is unclear to me why the homotopies witnessing that $\pi_2:\mathcal{C}\times \mathcal{M} \to \mathcal{M}$ is a homotopy equivalence should be $\Sigma$-equivariant. If so, I will certainly agree that any $E_{\infty}$ space is an $A_{\infty}$ space.

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  • $\begingroup$ You might want to mention that $\mathcal M$ is the "associative operad", the operad whose algebras are associative monoids. $\endgroup$ – André Henriques Nov 2 '14 at 21:00
  • $\begingroup$ Sure, I am sorry for that! $\endgroup$ – Edoardo Lanari Nov 2 '14 at 21:00
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    $\begingroup$ If $E_{\infty}$ and $A_{\infty}$ Operads are cofibrant replacements then the operad map $Ass\rightarrow Com$ gives you a map of operads $A_{\infty}\rightarrow E_{\infty}$, which in particular means than any $E_{\infty}$-algebra (space) is an $A_{\infty}$-algbera (space). $\endgroup$ – Ilias A. Nov 2 '14 at 21:01
  • $\begingroup$ @Fedotov: I can't understand your comment. Though I know quite well model categories, I have gone through operads only following May's concrete approach, hence I don't know any homotopy theory of operads yet. $\endgroup$ – Edoardo Lanari Nov 2 '14 at 21:04
  • $\begingroup$ Also asked on math.SE... $\endgroup$ – Najib Idrissi Nov 15 '14 at 10:15
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If $X$ and $Y$ are $G$-spaces for any group $G$, then the projection $\pi\colon X\times Y\to Y$ is a $G$-map, trivially: $\pi (gx,gy) = gy$. The map $\pi_2$ of your question is a very special case; note that $\mathcal M$ as I defined it is an operad as I defined operads, with $\Sigma_j$ acting on $\mathcal M(j)$.

Model categories are extremely important but entirely irrelevant here, and taking cofibrant approximations of operads tends to destroy their relevant individuality: different $E_{\infty}$ operads play seriously different roles in the applications, which are what people should care about.

Also, by my definition, $E_{\infty}$ operads are $\Sigma$-free rather than just locally contractible, so $Com$ is certainly not an example; $E_{\infty}$ spaces are far more general than $Com$ spaces, which of course are just commutative topological monoids.

This edit is to answer Lano's clarification of his question. Lemma 3.7, p. 24, op cit shows that a local equivalence over $\mathcal M$ is necessarily a local $\Sigma$-equivalence. The statement of Corollary 3.11 follows.

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  • $\begingroup$ Dear professor May, I have been imprecise. What is not clear is why are the homotopies (witnessing that $\pi_2$ is a homotopy equivalence) equivariant? $\mathcal{C}(j)$ is contractible, though it is explicitely stated that the homotopies involved are not necessarily equivariant. $\endgroup$ – Edoardo Lanari Nov 2 '14 at 23:03
  • $\begingroup$ Oh, I now see what I was missing! Thank you very much! And, by the way, I really appreciate that book! $\endgroup$ – Edoardo Lanari Nov 3 '14 at 0:15

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