Let $E$ be an operad in topological spaces. $E$ is usually called an $E\_{\infty}$-operad, if all the spaces $E\_n$ are contracticle. If $E$ acts on a space $X$, then by the recognition principle, $X$ turns out to be an infinite loop space. 



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> How much of the theory is preserved, if I replace *contractible* by *weakly contractible*? Do I still have deloopings, if $E$ acts on a space?
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Weak operad and deloopings
To answer your question and complete Justin's answer, one can pick a cofibrant replacement of $E$ to get a genuine $E\_\infty$-operad $F$ acting on the space $X$, and conclude that $X$ has an infinite delooping up to group completion issues.


But the other way round fails: not all infinite loop spaces are acted on by $E$. Example: any group like commutative monoid is an infinite loop space, but not all infinite loop spaces are commutative monoids (otherwise no non-trivial Dyer Lashof operation would exist).


Remark: If $E$ is weakly-contractible, then the constant maps $c: E\_n\rightarrow pt$ define an acyclic fibration, in the category of topological operads, from the operad $E$ towards the operad of commutative monoids $C$.
If you have a preferred (cofibrant) $E\_\infty$-operad $F$, then you can use the LLP to get an operad weak-equivalence $f: F\xrightarrow{\sim} E$, making $F$ a cofibrant replacement of $E$.


