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I was looking for particular and explicit examples of $E_\infty$-operads. I know the $E_\infty$-operad defined by Smith in http://arxiv.org/abs/math/0004003, and the Barratt-Eccles operad, but it is hard to find in the literature something concrete. Thanks!

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    $\begingroup$ I suspect that the meaning of "concrete" will depend on what you want your operad to act on. $\endgroup$ May 8, 2016 at 13:26
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    $\begingroup$ In that case, the most obvious one to study should be the Little $\infty$-cubes operad defined by May in "The Geometry of Infinite Loop Spaces". As the title suggests, this operad acts naturally on infinite loop spaces. $\endgroup$ May 8, 2016 at 13:38
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    $\begingroup$ Definitely the works of May in the early 70's is a good place to look. The linear isometries operad is another example. $\endgroup$
    – Todd Trimble
    May 8, 2016 at 14:29
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    $\begingroup$ I suggest that you take a look to Segal's paper Categories and cohomology theories. Although he does not use operads explicitely, I think it probably contains the best proof of the main theorems about the $E_\infty$-operad. After you've understood that you can look into May's The Geometry of infinite loop spaces and similar papers $\endgroup$ May 8, 2016 at 14:42
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    $\begingroup$ Please, Denis, that makes no sense: you don't learn about operads from a paper that never mentions them, and I could write for hours about the comparison of infinite loop space machines, which is not what the question is about. There are quite a few $E_{\infty}$ operads defined in a variety of categories. Just in spaces there are many others besides those already mentioned. A recent reference describing some of the virtues and defects and some of the roles of various examples is math.uchicago.edu/~may/PAPERS/Final1.pdf $\endgroup$
    – Peter May
    May 8, 2016 at 19:43

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