In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.

There exist $\Sigma$-free operads $\mathcal{C}_n$, $1\le n\le \infty$, such that every n-fold loop space is a $\mathcal{C}_n$-space and every connected $\mathcal{C}_n$-space has the weak homotopy type of an $n$-fold loop space.

For $\mathcal{C}_n$ the explicit example constructed in Peter May's book is the operad of little $n$-cubes.

However, there is a stronger version of the recognition theorem for grouplike $\mathcal{C}_n$-spaces spread all over the literature (e.g. at the nLab).

Although I'm absolutely not an expert in this field, I believe it should not be hard to prove the stronger form with the help of the techniques developed in May's book as soon as one dwelled on. (Is this the case? Is there a reason May did not state and prove the stronger version?)

Is there a reference for the stronger version that one can use if one does not want to expect the readers to be capable extending May's proof? I don't want to expect any knowledge of higher category theory, so Lurie's text on $E_k$-algebra does not fulfill my needs.

  • $\begingroup$ Adams, "Infinite loop spaces" seems to be the reference you are looking for. $\endgroup$
    – Oblomov
    Apr 8 '15 at 11:23
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    $\begingroup$ He does not prove it and references the book of May. $\endgroup$ Apr 8 '15 at 11:43
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    $\begingroup$ There is a Tex-ed version of the book on May's webpage, in which one can search for the term "group-like" (with the dash!). It occurs on p. 85 of the Tex-ed version, which I believe will answer your question in the case $n=\infty$. (I tried to cut and paste it as an answer, but the Tex got messed up and I don't have the time to edit it at the moment.) I think the same reasoning works for $n<\infty$. $\endgroup$
    – Dan Ramras
    Apr 8 '15 at 14:00
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    $\begingroup$ The first "trick" he mentions there ($X\simeq X_0\times\pi_0(X)$ and both factors are $n$-fold loop spaces) to get the result is not satisfying for me as a want the weak equivalence to be a map of $\mathcal{C}_n$-spaces. $\endgroup$ Apr 8 '15 at 14:21
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    $\begingroup$ The second comment of him how to obtain the stronger statement is basically that his proof can be adapted to the more general case, which I was aware of. I'm lacking a reference for exactly this generalization for readers that are not experienced with the work. $\endgroup$ Apr 8 '15 at 14:23

May gave a proof in the case $n = \infty$ in this followup paper to "Geometry of iterated loop spaces," which relies on knowing that a certain map (from a free algebra on $X$ to the free infinite loop space on $X$) is a group-completion map. On page 67 he asserted "I am reasonably certain that $\alpha_n: \mathcal C_n X \to \Omega^n \Sigma^n X$ is a group completion [...] but a rigorous calculation of $H_* \mathcal C_n X$ is not yet available." I believe that the lack of this calculation is the reason why the nonconnected case was not addressed.

The calculation happened in of Cohen-Lada-May's "The homology of iterated loop spaces." The group completion theorem is in part III, section 3, corollary 3.3 (page 226).


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