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In this question, Eric Zaslow asks about $E_\infty$ algebras. Here I ask the same question but for $E_n$ algebras. Can anyone give me a plain-and-simple definition of an $E_n$ algebra, which avoids the word "operad"? (A definition that does involve the word "operad" is: An $E_n$ algebra is an algebra over (chains on) the little $n$-discs operad.) For $E_1$ algebras, which are the same as $A_\infty$ algebras, we have a definition involving a bunch of maps $m_n$ satisfying some quadratic relations. Is there any analogous definition for $E_n$? I'd be interested in even just $E_2$ or $E_3$.

Edit: I have nothing against operads. As Charles Rezk points out, I'm just asking for a nice presentation of an $E_n$ operad which yields a nice definition of $E_n$ algebra involving some nice generators and some nice relations. I realize that "the reason" for the nice definition of $A_\infty$ algebra is because of the Stasheff associahedra. So I wonder if there are any analogous things for $E_n$? Or, if such a nice presentation is not possible or not feasible, then why not?

(Assume characteristic zero if needed.)

edited Nov 17 2010 at 19:41

asked Nov 17 2010 at 18:04

Kevin Lin
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Having "a definition involving a bunch of maps $m_n$ satisfying some quadratic relations" means you've got a presentation of some flavor of $A_\infty$-operad in terms of generators and relations. So you are asking for an explicit presentation of an $E_n$-operad, but one which never mentions the word "operad". Is that right? – Charles Rezk Nov 17 2010 at 18:59

Correct. . – Kevin Lin Nov 17 2010 at 19:07

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I claim my comment to Eric's question is an answer for yours. The "sequence operad" of McClure-Smith is an entirely explicit presentation of an $E_\infty$-operad (you'll have to decide if it's "nice"). In that paper, they describe explicit suboperads of the sequence operad which are $E_n$-operads. – Charles Rezk Nov 17 2010 at 19:50

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@Charles: Ah, thank you. I'll look at those papers. – Kevin Lin Nov 17 2010 at 19:55

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If characteristic zero is allowed, then you could get a different answer from the article: ambp.cedram.org/item?id=AMBP_2004__11_1_95_0 In this reference, the structure of a homotopy $n$-Gerstenhaber algebra is given in terms of mappings $m_{n_1,\dots,n_r}: A^{\otimes n_1+\dots+n_r}\rightarrow A$, satisfying symmetry relations and some boundary equation as in the $A_\infty$-case. The $n$-Gerstenhaber operad, defined as the homology of the little $n$-cubes operad, is weakly-equivalent to the chain operad of little $n$-cubes in characteristic zero (formality theorem), and hence is $E_n$. – Benoit Fresse Nov 17 2010 at 21:00

Definition of an E_n algebra

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