In [Geometry of the iterated loop space](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.170.2840&rep=rep1&type=pdf), Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.



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> 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.
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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](http://ncatlab.org/nlab/show/little+cubes+operad)).


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](http://arxiv.org/abs/0911.0018) does not fulfill my needs.



Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces
May gave a proof in the case $n = \infty$ in [this followup paper](http://www.math.uchicago.edu/~may/PAPERS/13.pdf) 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](http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf)." The group completion theorem is in part III, section 3, corollary 3.3 (page 226).


