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Which is the exact original formulation of May recognition theorem? At Ek-Algebras Lurie’s general forumlation is given and it is said that for the $\infty$-topos being $Top$ this reduces to May’s theorem. But it seems to me that even for $Top$ Lurie’s formulation tells much more than May’s theorem as it is stated there. namely in Lurie’s formulation we have an equivalence, which is missing in the formulation of May’s theorem given there. Moreover the equivalence in Lurie’s verison is an equivalence of $\infty$-categories and I would not be able to say that in a pre-higher categories languages (maybe it is an equivalence of homotopy categories in the original version?). Another thing puzzling me is that if $X$ is homotopy equivalent to an $n$-fold loop space I don’t see any natural action of the little $n$-cubes operad on $X$, but only an action up to coherent homotopy. And again, to speak of actions up to coherent homotopy it seems I need a language of higher operads (but I may be wrong here).
Which is the exact original formulation of May recognition theorem?
See in
First version there is theorem 1.3, which goes
There exist $\Sigma$-free operads $\mathcal{C}_n$, $1 \leq n \leq \infty$, 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.
We should add more details about all this to the $n$Lab entries. But I won’t do it tonight :-)
Never mind.
Thanks a lot for the comments and the answers! I’m still a bit puzzled by the homotopy invriance issue. I mean, I’d like to have a statement of the form "A (nice) topolological space has the homotopy type of an $n$-fold loop space if and only if…" where "(nice)" includes all the needed features like "grouplike" or "$(n-1)$-connected" for this statement to be true. But I don’t clearly see what should be in place of the "…" in the statement.
Namely, I know the following (in the notations from post #4 above), always assuming $X$ to be as nice as needed:
i) if $X$ is homeomorphic to an $n$-fold loop space, then there is a $C_n$-action on $X$;
ii) if there is a $C_n$-action on $X$, then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space;
iii) if $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space then there is a $WC_n$ action on $X$.
how does this continues? I’d like the next step stabilizes, i.e.,
iv) if there is a $WC_n$ action on $X$ then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space
but I do not know whether this is true or not
Answer to last question iv): Yes it is true, by my original arguments as cited in my post 4. The operad WC_n maps by an equivalence to the operad C_n. That is all that is needed to carry out the argument: I crossed C_n with an E_{\infty}$ operad in many of my applications and exploited the resulting map to C_n. You exploit the map from WC_n in exactly the same way.
I see: so the idea is to replace the projection $\pi_n:\mathcal{C}\times \mathcal{C}_n\to \mathcal{C}_n$ used in your “The Geometry of Iterated Loop Spaces” with any cofibrant resolution $\phi_n:\mathcal{Q}_n\to \mathcal{C}_n$ (in particular one can use Boardman-Vogt $\mathcal{W}\mathcal{C}_n\to \mathcal{C}_n$) and then verbatim follow your argument. Thanks a lot!
@jim - I asked a stupid question, then realised it was stupid rather quickly, so deleted it.
Re #11: Why not add a comment about this to the article?
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