One possible answer: Stasheff proved that a (connected) space $X$ is (homotopy equivalent to) a loopspace if and only if $X$ is an algebra over the $A_\infty$ operad (or rather I should say *an* $A_\infty$ operad).

See for instance [this article][1].

[1]:http://www.ams.org/notices/200406/what-is.pdf