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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 2010 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 2010 at 2:03
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 2010 at 10:35

There's a lot of things you could ask.

• 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?

• (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 2010 at 5:29
Every operad fits into a prop. – Todd Trimble Feb 27 at 21:42

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.

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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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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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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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 2010 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 2010 at 6:06
Anything that started in the 70's is pretty young. I mean, I started in the 70's, therefore... ! – Ryan Budney Aug 21 2010 at 6:39
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 2010 at 19:47
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 at 5:09
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Here are a few of my favorite questions about operads at the moment:

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