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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?



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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. – André Henriques Nov 25 '10 at 17:38

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