What are natural questions to ask about an operad?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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 https://arxiv.org/abs/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.

Answer 4

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!

Answer 5

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).