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Hello,

The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is:

Let $X$ be a (group-like) topological space acted on by the little $n$-discs operad, then $X$ is (weakly) homotopy equivalent (as an algebra over the little discs operad) to an $n$-fold loop space.

In particular, the theorem leaves me unsatisfied since there are some spaces which have the homotopy type of an $n$-fold loop space but which are not acted on by the little discs operad.

The "right" statement of the recognition principle should be a statement inside the homotopy category (of spaces and operads), but I have never seen it stated properly.

Does it appear somewhere? Also, is there a modern version of P. May's proof that appears in the geometry of iterated loop spaces?

Thanks,

K.

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    $\begingroup$ In "the geometry of iterated loop spaces", the main theorem is stated in a way that allows for operads that are not necessarily isomorphic to the little n-cubes operad, but only homotopy equivalent to it. $\endgroup$
    – André Henriques
    Nov 25, 2010 at 17:38
  • $\begingroup$ sorry for digging out this old question but one aspect in your comment I not understand. What is a homotopy equivalence between operads? $\endgroup$
    – user267839
    Jul 20, 2020 at 0:38

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