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I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the structure of a group, I'd ask if it is abelian or has torsion, etc. So what are these questions for operads?

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This is not my subject but the standard question I have heard asked is "Is it Koszul?" –  Bruce Westbury Aug 20 '10 at 21:27
    
I think that the answer strongly depends on the category: linear operads (in dg-Vect) and topological operads (in TOP) have little in common. –  Victor Protsak Aug 21 '10 at 2:03
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Concerning operads in dg-Vect vs Top, the generators and relations picture used often with the former obviously doesn't translate very well to topological operads, but really quite a lot of the homotopy theory is essentially the same in the two categories. E.g., (co)bar constructions and cofibrant replacements are really fundamentally homotopy theoretic and they should make sense for operads in any model category. –  Jeffrey Giansiracusa Aug 21 '10 at 10:35
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There's a lot of things you could ask.

  • Operads act on things, that's their point. What things does your operad act on? Presumably this is how you found your operad. Moreover, once you know it acts on something you can ask if that action is maximal, whether or not your operad fits into a bigger operad that also acts on the thing in question, etc. Similarly, you can ask what does that operad tell you about the thing its acting on.

  • Operads can have sub-operads, do you have any interesting ones? That would lead to other related questions, like is your operad an extension of other operads? Take a look at the Markl and Stasheff operad book to get a sense for some of the operads out in the literature, and what they're good for.

  • There are things like totalizations and bar constructions for operads. What might that look like for your operad?

  • Operads induce other operads, for example, the homology of a topological operad is another operad. Does your operad have any related operads that are known or otherwise interesting?

  • (Edit, idea from Jeff's cyclicity suggestion) Operads sometimes fit into even larger higher-algebraic structures. Jeff mentions cyclic operads, but there are also PROPs, for example. You might want to consider that maybe you're dealing with something that's "more than" an operad.

That'd be a start.

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To elaborate on the first point, which would be my first answer too, one thing which is nice to know is whether algebras over an operad (that is, things the operad acts on or equivalently objects in your category whose endomorphism operad receives a map from the operad in question) admit a different, possibly familiar, description. For example, one could "discover" the Lie operad (studying the Goodwillie tower of the identity functor for example) and then realize that algebras over that operad are of course quite familiar. –  Dev Sinha Aug 21 '10 at 5:29
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Every operad fits into a prop. –  Todd Trimble Feb 27 '13 at 21:42
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Here are a few of my favorite questions about operads at the moment:

  1. Is your operad actually a cyclic operad (in the sense of Getzler-Kapranov)?

  2. If so, what is the modular operad that it generates?

  3. Is it a quadratic operad? Is it Koszul?

  4. Is it cofibrant? If not, what does a nice cofibrant replacement look like? E.g., the associative operad is not cofibrant, but the A-infinity operad made from the associahedra is an interesting cofibrant replacement.

  5. Does it have interesting morphisms to or from other operads? If so, then people are often interested in the deformation theory of such a morphism.

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(4) is absolutely my favorite question to ask about operads. There are tons of papers about the connection between model categories and operads –  David White Jan 29 '13 at 22:16
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Hi Connie. Let me use your question as an excuse for an extended answer. A pair of brief papers "Definitions: operads, algebras and modules" and "Operads, algebras and modules", which are available at http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf and http://www.math.uchicago.edu/~may/PAPERS/handout.pdf (# 84,85 on my website) give several variants and reformulations of the original definition together with some history of antecedents, a variety of algebraic and topological examples, and the crucial relationship with monads that led me to coin the word "operad". There is also a discussion of the relationship to homological algebra, showing how the homological theory simplifies if you work over a field of characteristic zero and, in contrast, how operads encode homology operations (Steenrod operations and Dyer-Lashof operation) if you work over a field of finite characteristic. Notes for a talk, http://www.math.uchicago.edu/~may/TALKS/SwitzerlandTalk.pdf, expand on the last point.

The distinction of characteristic illustrates a general point. Operads are defined in any symmetric monoidal category, and the right questions to ask depend in large part on what category you are working in. It may make no sense at all to ask algebraic questions of a topological operad or topological questions of an algebraic operad. There is also a distinction to be made about questions to ask about operads and questions to ask about their algebras. Incidentally, groups are by design not examples of algebras over an operad: to define inverses, you need diagonals, and operads are not intended, or rather intended not, to incorporate such structure. The questions to ask also depend on what role your observation plays. Operads allow a taxonomy of certain types of algebraic structures, so the question may just be "what kind of structure am I looking at".

But you might also want to ask whether the algebras you are looking at give simpler "approximations'' of more complicated or less accessible structures that occur "in nature". For example, spaces $\Omega^n\Sigma^n X$ occur in nature, but they can very usefully be approximated by the monads $C_nX$ associated to appropriate operads.

You might also want to ask if operads can be used to define rigorously new structures that you want to study. A very recent example arose in work of Bertrand Guillou and myself in equivariant infinite loop space theory: there is an intuition of what a genuine strict symmetric monoidal $G$-category should be, one that gives rise to a genuine $G$-spectrum; the best definition we know is that such a category is an algebra over a particular operad in $Cat$ (see http://front.math.ucdavis.edu/1207.3459). Quite a few recent variants of the definition of an operad arose analogously.

In algebra, very simple operads prescribe very natural and previously unstudied kinds of algebras. Loday and some of his students (I'm blanking on names) gave a number of examples.

While one can ask questions about the homotopy theory of operads in general, using model category theory, that is perhaps my least favorite question to ask: it rarely cuts to the heart of the applications, excepting those in higher category theory, or so it seems to me. Model categories of algebras over particular operas do play a major role in many applications, albeit sometimes only implicitly.

I'll stop here, since I could go on forever.

One comment. While the Martin-Shnider-Stasheff book is a useful compendium, its treatments of different topics are not all at the same level, and you might well prefer less comprehensive treatments that better address your directions of interest. And people should be warned that the definition of an operad in that book is actually incorrect: it omits a crucial equivariance property that is of real importance in applications. For example, it plays a key role in the proof of the Adem relations for the Steenrod and Dyer-Lashof operations. Benoit Fresse's book "Modules over operads and functors" gives a quite different take on operads, with a focus on modules over algebras over operads.

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I think this might be worthy of being a separate answer, so here I go.

May has two versions of Geometry of Iterated Loop Spaces on his website, one type set and one in Tex. This is where operads were invented (even if in name only, I won't stress about history, and I mean no disrespect). It is helpful if only because it is an early treatment, do not read it too long, probably not past chapter 5. The diagrams will not seem natural until you have your two examples, the little n-cubes operad and the endomorphism operad of a topological space.

Then, if not before, read a little bit of Adams Infinite Loop Spaces, he uses props but thats ok.

These are just little introductions stressing the homotopy theory side, there are other aspects, I just dont know them (I barely know the homotopy theory side). And when you get sad and stuck on something, just look up some new operad, here are two examples: the swiss cheese operad and the cactus operad.

thanks Ryan for reminding me of May, almost everything he has published is legally available on his website, if not more!

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If you're interested in odd variants of the cubes operad, I'd be curious what you think of what I'm calling "the overlapping cubes/discs/framed discs" operad. The operad of overlapping n-cubes is equivalent to the operad of n+1-cubes, but it's a multiplicative operad (containing the associative operad). And the definition is quite elementary, like the cubes operad. This is in my preprint called "an operad for splicing" on the arXiv. The operad of overlapping intervals "feels" quite a bit like the cactus operad. –  Ryan Budney Aug 26 '10 at 5:19
    
ah, but does your overlapping interval operad have a circle action lurking? (doesn't the Cactus operad? or some sort of circle operading action thingy) –  Sean Tilson Aug 26 '10 at 5:46
    
It looks like the cactus operad in another way. :) I'm curious what people make of this, and if anyone has seen this operad before. I'll make it a new question. –  Ryan Budney Aug 26 '10 at 5:51
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From a primarily combinatorial point of view of operads, here are a list of typical questions that you can ask about an operad $\mathcal{P}$.

  • If $\mathcal{P}$ is defined as a suboperad of a bigger operad $\mathcal{Q}$ and have finitely many elements (or finite dimension) of arity $n$ for all $n \geq 1$, you can ask about its dimensions and its Hilbert series;

  • You also can ask about a presentation of $\mathcal{P}$ by generators and relations between its generators;

  • If $\mathcal{P}$ is defined by a presentation you can ask about a realization of $\mathcal{P}$, that is an explicit example of an operad that admits same presentation as $\mathcal{P}$ (thus, in other words, an operad isomorphic with $\mathcal{P}$) and a way to compute explicitly the composition of two objects;

  • It is also worthwhile to study the symmetries of $\mathcal{P}$ and find a precise description of its automorphisms and antiautomorphisms;

  • And of course, it is always interesting to describe and study the suboperads of $\mathcal{P}$ (previous point can help for this one).

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Just about any basic question you can ask about operads is asked and answered in the treatise by James Stasheff,Martin Markl and Steven Shnider.I really don't think anyone can give your question a better answer then that.James is particularly proud of the work that lead to the book,from what he told me.

Of course,the really important question is whether or not you can discover anything about operads that's NOT in the book,Connie.........

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Operads are a young subject and there's plenty known about operads that isn't in the MSS book, in large part because many things have happened since the book was printed. Take a look at Giansiracusa's papers, for example. –  Ryan Budney Aug 21 '10 at 5:13
    
I agree that not everything is in Stasheff-Markl-Shnider book, but disagree that "operads are a young subject". For example, the book of Boardman and Vogt was published in the early 1970s. It would probably be more accurate to speak of several stages in the development of the operad theory. –  Victor Protsak Aug 21 '10 at 6:06
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Anything that started in the 70's is pretty young. I mean, I started in the 70's, therefore... ! –  Ryan Budney Aug 21 '10 at 6:39
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It is a nice book but for all that I think it would take a pretty particular student for that to be the best first-point-of-exposure. For a first exposure to operads I'd probably choose a very concrete example. The cubes operad, the recognition principal for iterated loop spaces, and then the general definition of an operad, like in these notes: math.uchicago.edu/~may/PAPERS/handout.pdf IMO that would be a solid, generic starting-point. After that, MSS would be a fine 2nd step. –  Ryan Budney Aug 23 '10 at 19:47
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Just reading this now. I think Ryan is on target here: the early works on the little cubes etc., contained for example in The Geometry of Iterated Loop Spaces by May, are an excellent and non-trivial way to see operads at work. From there, tastes can dictate where to go next. (My own tastes lead me to contemplate the extreme generality and fundamental importance of operads, as witnessed for example in Tom Leinster's Higher Operads, Higher Categories, but the works of May will always occupy a fond place in my heart. (-: ) –  Todd Trimble Feb 28 '13 at 5:09
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