As a non expert in the theory of topological operads, I find it pretty hard, to understand what algebras for little balls/cubes/something operads are.

For all the other famous operads I know (like Lie, Com, Ass ect.) associated algebras are meanwhile given in terms of generators and relations, which makes them understandable without any reference to operads. However this seams to be different for the little something operads.

I know that their homology is sometimes a Poisson-n algebra for $n\geq 2$, therefore algebras for the homology are (homotopy) Poisson-n algebras in that case. But that's just the homology.

Then I heard, that in a category those algebras are just commutative algebras, as long as $n\geq 2$. Is that necessarily true?

So my questions are:

1.) Are there generator&relation style descriptions of the algebras for the little n-balls/n-cubes or related operads?

2.) Do I need a higher category to get examples that are not just commutative for $n\geq 2$?

3.) Is there a standard reference for those algebras? (Not just the operads)


4.) How could I start to get a good understanding of those algebras? Assuming I know just the basic about the underlying operads.

  • $\begingroup$ 3 no's :) I don't have a proper computer at hand now, otherwise I could contribute to the debate that will probably start now. Maybe later. $\endgroup$ – Fernando Muro Jun 3 '15 at 14:11
  • $\begingroup$ Ok I understand, you can't contribute right now, but for everyone else a little more background would be appreciated ;-) $\endgroup$ – Mark.Neuhaus Jun 3 '15 at 14:14
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    $\begingroup$ Algebras for these operads are, morally speaking/modulo some hypothesis, n-fold loop spaces. This is an important theorem. There is more than just a Poisson structure on their homology (unless you are working in characteristic 0). Fred Cohen describes their homology in "homology of iterated loop spaces" which can be found on Peter May's website. I don't think you will ever find a satisfactory answer to #1. Eric Zaslow asked a similar question a long time ago. $\endgroup$ – Sean Tilson Jun 3 '15 at 15:20
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    $\begingroup$ I am aware of this, but the question doesn't mention $E_n$, it mentions specific models of an $E_n$-operad. This is part of the reason I didn't give an answer. Sheesh. $\endgroup$ – Sean Tilson Jun 3 '15 at 17:29
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    $\begingroup$ You asked, or at least it seemed to me, about a specific point set model of an $E_n$-operad and did not specifically mention $E_n$ in your question. Qiaochu, as he demonstrates in his answer, is talking about something more general. $\endgroup$ – Sean Tilson Jun 4 '15 at 7:11

An $E_n$ algebra (an algebra over the little $n$-cubes operad, etc.) is intuitively an object with $n$ compatible monoid structures. All of the subtlety in this theory lies in making "compatible" precise; in particular it is not a property but a structure.

Here are some examples.

  • In $\text{Set}$, an $E_1$ algebra is a monoid. For $n \ge 2$ an $E_n$ algebra is a commutative monoid by the Eckmann-Hilton argument.
  • In $\text{Ab}$, an $E_1$ algebra is a ring. For $n \ge 2$ an $E_n$ algebra is a commutative ring by the Eckmann-Hilton argument.
  • In $\text{Top}$, a grouplike $E_n$ algebra (an $E_n$ algebra where $\pi_0$ is a group) is an $n$-fold loop space $\Omega^n X$ by the recognition principle. The $n$ monoid structures are the loop compositions in the $n$ loop directions. A grouplike $E_{\infty}$ algebra is an infinite loop space, or equivalently a connective spectrum.
  • In $\text{Cat}$, an $E_1$ algebra is a monoidal category, an $E_2$ algebra is a braided monoidal category, and for $n \ge 3$ an $E_n$ algebra is a symmetric monoidal category. (This stabilization phenomenon is related to Freudenthal suspension and is part of the "periodic table of higher categories.")

In more detail, let's focus on $n = 2$. An $E_2$ algebra is intuitively an object with two compatible monoid structures. The Eckmann-Hilton argument shows that they're equivalent, but the way in which it shows that they're equivalent is itself interesting structure: along the way, it describes a map between $ab$ and $ba$ (for both monoidal structures), which is a braiding. This is where braided monoidal categories come into the picture. It might help to stare at the standard proof of Eckmann-Hilton where you move squares around and to explicitly think of the squares as describing binary operations in the little $2$-cubes operad.

One way to make "compatible" precise is to write down a presentation of the $E_2$ operad, which is unique among the $E_n$ operads ($n \ge 2$) in that all of its spaces are $1$-truncated: that is, they are all groupoids, or equivalently have no higher homotopy $\pi_n, n \ge 2$. There is a particularly nice model of the $E_2$ operad as an operad in groupoids called the parenthesized braid operad, which has a "generators-and-relations" presentation, but where it's important to understand that when specifying an operad in groupoids there are three sorts of things you might want to write down, rather than two:

  • Operations (which then generate other operations under operadic composition),
  • $1$-morphisms between operations (to describe the groupoid structure), and
  • Relations between $1$-morphisms between operations (to further describe the groupoid structure).

In general, higher category theory blurs the distinction between generators and relations: relations become generators one categorical level up. To avoid having to make the distinction you can just say "presentation."

The standard presentation of the parenthesized braid operad mimics exactly the standard axiomatization of braided monoidal categories: there is a generating binary operation (the monoidal structure), two generating $1$-morphisms (the associator and the braiding), and some relations between these (the pentagon and hexagon axioms). Here I'm ignoring units for simplicity. This presentation is important in discussions of Grothendieck-Teichmüller theory.

For $n \ge 3$ the spaces in the $E_n$ operads aren't truncated: for example the space of binary operations is the sphere $S^{n-1}$, which has nontrivial homotopy groups in arbitrarily high degrees. So I don't think there's any hope for a presentation along the above lines in general. You could ask for presentations of various truncations, but I imagine these are pretty horrible to work with in general. The $E_n$ operads exist precisely so that you can avoid having to do stuff like this. Of course it's a different story after taking rational homology.

$E_n$ algebras show up in the story of factorization homology and topological field theory, so that's one place to go for some resources; see, for example, these notes by Ginot.

  • $\begingroup$ Qiaochu, I tink that is a good answer, but in parallel I think I miss a lot of background to make all the connections. Can you say a bit on how to achieve this knowledge in detail? Is there a roadmap of papers or something to read? Luries Higher algebra, maybe, but avoiding the machinery of quasie categories would be nice. $\endgroup$ – Mark.Neuhaus Jun 3 '15 at 18:00
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    $\begingroup$ Intuition here is fine, but what, precisely, do you mean by an "E_n operad" in a general context. I know precisely what we mean in topology (or differential graded module) contexts. $\endgroup$ – Peter May Jun 3 '15 at 23:48
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    $\begingroup$ @Peter: you probably won't like this, but there is a unique (up to isomorphism) (symmetric) $\infty$-operad that I mean when I say "the $E_n$ operad," which has various cofibrant models in various model categories of topological operads, and I can talk about its $\infty$-algebras in any symmetric monoidal $\infty$-category. I don't need a different operad for every occasion: to tell you where the $n$-ary operations go I just need a space of $n$-ary operations and an endomorphism space $\text{End}(X^{\otimes n}, X)$, and a map of spaces between them, plus coherences. $\endgroup$ – Qiaochu Yuan Jun 4 '15 at 5:16
  • $\begingroup$ In the $n = 1$ case and with no $\infty$s what I'm saying is that I can tell you what a monoid in $(\text{Vect}, \otimes)$ is knowing only what the $E_1$ operad looks like as an operad in sets, without having to write down a new operad for the occasion. Of course if I wanted to I could also tell you what a monoid in $(\text{Vect}, \otimes)$ is by writing down an operad in $\text{Vect}$, namely the one given by taking free vector spaces on my first one, and that might be a good idea for various reasons. $\endgroup$ – Qiaochu Yuan Jun 4 '15 at 5:19
  • $\begingroup$ @Mark: I certainly can't claim to have detailed background knowledge here; at best I have some heuristic pictures that I'm reasonably comfortable with. If you don't know much homotopy theory it may be helpful to strengthen your background there. Other than that, it would be helpful if you had a more specific request. $\endgroup$ – Qiaochu Yuan Jun 4 '15 at 5:27

May I suggest you the following monographs: " Homotopy of Operads and Grothendieck-Teichmüller Groups" by Benoit Fresse available here: http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/ together with video recordings of a master degree course on that subject http://math.univ-lille1.fr/~operads/2012courses.html#Lille

  • $\begingroup$ Those links look like a lot of stuff to get through. So what you say is that these courses will lead to an understanding of algebras for these top. operads? $\endgroup$ – Mark.Neuhaus Jun 3 '15 at 14:47
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    $\begingroup$ In particular, you should look at this chapter: math.univ-lille1.fr/~fresse/OperadHomotopyBook/EnOperads.pdf $\endgroup$ – David C Jun 3 '15 at 14:52
  • $\begingroup$ That latter paper you referenced, was preisely what I allrady knew. Unfortunately it only talks about iterated loop spaces and not the various n-algebras that appear elsewhere in the context of higher categories. $\endgroup$ – Mark.Neuhaus Jun 3 '15 at 16:01
  • $\begingroup$ the videos don't work $\endgroup$ – john mangual Jun 3 '15 at 17:40
  • $\begingroup$ @johnmangual It seems that only the first video isn't working, the other ones work fine (at least for me). (Though I don't know how feasible it is to follow the course without the first lecture...) $\endgroup$ – Najib Idrissi Jun 4 '15 at 7:57

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