Why are operads useful?

Question

The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar graphs and so one can use planar geometry (say, the Jordan lemma) to investigate the word problem in complicated groups. Similarly, asymptotic cones are useful in geometric group theory because they allow to study large scale properties of a discrete object (a group) by looking at small scale properties of a continuous object. I would like to know a similar answer for operads.

Update Many thanks to everybody for your answers. Unfortunately I can accept only one. So I just accept the first answer.

Answer 1

Here are a couple 2-3 line answers to your question:

1) They allow you to treat various algebraic problems uniformly. For example, Commutative, Associative, and Lie algebras all have their own cohomology theories (Harrison, Hochschild, and Chevalley-Eilenberg respectively). These can all be seen as instances of a single operad cohomology.

2) One can use operads to construct cohomology classes for the Mapping Class group and Out(Fn) (and others). The idea is to use graphs "colored" by operads, and construct a chain complex out of these colored graphs that computes the desired cohomology.

and perhaps the most classic answer:

3) They allow you to classify loop spaces and infinite loop spaces. For connected spaces, these are exactly classified as algebras over various operads.

Answer 2

Operads come into play whenever you're dealing with a family of objects that have coherent operations. An inaccurate way to get it across but which gets quite close to the spirit would be to think about things like computing the number of ways of partitioning $10^{100}$ into $60$ subsets, vs. the generating function for partitions. Perhaps the generating function does not help you compute this one instance of the partition problem, but it does inform on some general aspects of the partition problem, such as asymptotics.

To make the analogy more complete, operads are relevant when you're dealing with things like:

• A universal algebra. This is in some sense the original operad idea -- operads were designed to be a category-independent notion of universal algebra.

• Whenever you have families of spaces that have families of ways in which they can be combined. Topological operads were designed to encode this kind of information. The first concrete instance of this was the cubes operad, which acts on iterated loop spaces. Cubes operads have the added benefit that they can be used to deduce that spaces are iterated loop spaces.

In that regard operads tend not to be used to completely plumb the depths of one particular object. They tend to be used to extract coherent information about a family of objects (unless of course your object is itself a "family of objects").

I think in group theory the way this kind of thing comes up would be when you take classifying spaces. For example, pure braid groups. There are various maps $P_n \times P_k \to P_{n+k-1}$ given by "blowing up" one of the strands of the $n$-stranded braid and inserting the $k$-stranded braid in its place. When you take classifying spaces, these are the structure maps for the operad of $2$-cubes. I suppose it's subjective, but the homology and cohomology of the pure braid groups have a far more pleasant exposition in this operadic framework -- and it's quite analogous to the initial generating function analogy -- than say in the language of explicit group cycles and cocycles.

So I suppose my point could be phrased as operads do not do anything, in some computational sense. They just hold information in a more pleasant way. They're more akin to a data type than an algorithm to perform a task.

Answer 3

I find two different points of view useful.

1. Further to Steve's first answer, I would say that operads put many algebraic structures into one compact and useful meta-algebraic setting. Lie, associative, commutative, Poisson, Gerstenhaber, etc. All of these fit into one nice framework which then tells us how to define cohomology theories and study the deformation theory in each setting. This universal setup also tells us how to study generators and relations, homological algebra, duality theory, and so on. Operads, somewhat like category theory, allow one to see the common structure behind many a priori different worlds.

2. My other point of view is that operads, along with their siblings, the cyclic and modular operads, are all about studying structures that glue/compose along trees or graphs. Manifestations of this type of composition appear in topological field theory, infinite loop space theory, low dimensional topology, and all sorts of other places.

Answer 4

One reason that operads are used is in obstruction theory. Suppose we have a CW-complex $X$ with basepoint and multiplication $\mu:X \times X \to X$ which is associative and unital up to homotopy, and we want to know if $X$ is homotopy equivalent to a topological monoid $X'$ by a map that, up to homotopy, respects the multiplication. This is a question about homotopy theory, but constructing a strictly associative multiplication is not amenable to methods of homotopy theory.

This is a question about the associative operad, but we can replace it by a question about an equivalent operad such as the collection of Stasheff associahedra. This has a simple presentation, as an operad, in terms of generators and relations, and so it is easier to classify actions on an object. This provides a sequence of obstructions in $\pi_k Map(X^{k+3}, X)$ to finding on an object (and the choices are similarly parameterized).

Of course, as with many things in algebraic topology, this general method works far better to show something does not admit a multiplication, or when the obstructions occur in zero groups. However, it is difficult to attack such problems outside specific circumstances without using operads. (Perhaps someone else knows of methods that don't implicitly use operads; I don't.)

I'm not complete clear on what constitutes the difference between "where" and "why" in your statement, but I hope this qualifies.

Answer 5

There surely are many answers to this question... For me, one of the key reasons is that there are lots of situations where existing geometric and algebraic structures exhibit some kind of associativity. (Geometrically, think of gluing pants, like in Jeff's comment, algebraically think of composing operations and/or cooperations.) The notion of an operad allows to formalise this observation, and treat objects like that as associative algebras in a certain monoidal category, - and since we know an awful lot of ways to approach usual associative algebras, this gives intuition of how to approach problems for these, more tricky objects. I think the analogy with your examples is quite clear.

Answer 6

I'm not an expert by the way I could give you an answer based on my personal experience in my study of category theory.

Operads allow to make lots of constructions and to encode information of many mathematical object into an algebraic structure, this algebraic structure allows to work in a simpler way with the above mentioned objects.

In pratical I think operads are similar to homotopy/homology groups, which encode homotopical information of topological spaces and enable to distinguish such spaces in a simple way, studying algebraic structures. This is useful because is more simple classifying groups rather then topological spaces.

More in general I think the usefulness of all such structures derive by the fact that usually (concrete) algebraic structures are easier to work with, but I emphasise that these are just my thoughts.

Answer 7

I don't know the answer myself, but the book linked to (much of which is available on google books) purports to give several answers:

Martin Markl, Steve Shnider, Jim Stasheff (2002). Operads in Algebra, Topology and Physics. American Mathematical Society. ISBN 0821843621.