The idea of an operad was born in an attempt to understand when a space has the homotopy type of a loop space, or more generally an -fold loop space. An operad
is supposed to be a collection of operations with different arities with rules about how to compose them. To give an algebra over an operad
is to give a space
together with an interpretation of each of these “operations” as actual operations on
.
Motivation
The concrete motivation is as follows. Consider a pointed space , the loop space
is something that you want to think of as approximating a topological monoid. This is because you can compose paths. But the composition is only associative up to homotopy. However, essentially any way of composing a collection of
paths is equivalent to another way (i.e., differently parenthesized) of composing that collection of
paths, in that order, and that equivalence is canonical (up to homotopy, at least).
This is a somewhat long-winded way of saying that there is a multiplication law on which is not only homotopy associative, but homotopy coherently associative. What does that mean?
The best analogy I can come up with is from ordinary category theory. Consider the definition of a monoidal category. In a monoidal category , one usually does not want to require an equality of functors
; this is considered “evil.” Also, this does not generally happen in practice. Rather, one wants a canonical isomorphism between the functors
But this alone is not enough. Given such an isomorphism, if you have objects and tensor them in two different parenthesizations, but in the same order, applying the associativity isomorphism means that the two large tensor products will be the isomorphic. But there are many different ways to apply the associativity isomorphisms! Thus, in general, one might get multiple isomorphisms between these two differently parenthsized large tensor products. The coherence condition on the associator is that these two isomorphisms have to be equal. For instance, there are two ways of getting from
to
using associator isomorphisms; we want these two to be the same. This is MacLane’s pentagonal diagram.
Now, in homotopy theory, when we have a homotopy associative H space, the homotopies giving the homotopy between differently indexed products should be canonical, at least up to homotopy. This is the sort of thing that happens with the loop space : it is a homotopy coherently associative H space. It is much like a monoidal category in that sense. (A monoidal category in fact gives you a homotopy coherently associative H space by taking the nerve.)
So what does this all mean? Right now I’ve been very vague and hand-wavy. The definition of homotopy coherent associativity could be written out with explicit diagrams, using for instance Stasheff associhedra. One of the themes of modern homotopy theory, though, is to find clever ways of hiding homotopy coherence, because writing out massive explicit diagrams quickly becomes quite difficult. This is, for instance, one of the cornerstones of (and motivations for) many of the modern approaches to -categories.
Definition
OK, so what should an operad be? As before, it should be a collection of abstract “operations” together with laws how to compose them. In this way, many algebraic structures (such as groups and rings, but not fields) will become algebras over an operad. Let’s do operads in sets, for now.
An operad is going to consist of sets
of “
-ary operations” (which you should think of as maps
for some set
) with the following data:
- There is a composition law
. This corresponds to the process of composing operations.
- There is an action of the symmetric group
on
.
- There is a distinguished “identity element” in
corresponding to the identity function.
- There are standard associativity, unital, and symmetry constraints that one figures out from how composition of ordinary
-ary operations behaves.
The basic example is the operad of all -ary operations (for
varying!) on a set
, called the endomorphism operad. Given
, we can say that
acts on
if we are given a homomorphism of operads (which I won’t define) from
to the endomorphism operad of
. There is thus a category of sets with an
-action, which is called the category of
-algebras.
The definition of an operad (and that of an algebra), however, makes sense in any symmetric monoidal category. For instance, we can talk about operads of spaces, operads of chain complexes, operads of simplicial sets, operads of spectra, and so on.
In the topological case, the idea is that for an operad , the connectivity of the spaces
is a way of expressing homotopy coherence. For instance, let’s suppose the following. Each
has
equal to the symmetric group
, i.e. there are
components of
permuted by the action of
. Suppose all these components, moreover, are contractible, and that the composition law corresponds to composition in the symmetric group upon taking connected components. (In other words:
is the operad described below, the associative operad.)
Definition 1 Such an operad
is called an
-operad.
A simple example of an -operad is the associative operad. In this case, each
is just the discrete space
. Composition is defined in the natural way, via
where the first term controls the order of composition. In other words, there are
-ary operations, except that all are just distinct permutations of each other. So there is really one fundamental operation, in
, and everything else comes from there. One can check that an algebra over this operad is simply a topological monoid.
So an -operad is supposed to be a “homotopy coherent thickening” of the associative operad. For instance, the loop space
is almost never an algebra over the associative operad, i.e. it is not a monoid. But it is a homotopy coherent monoid in the sense that it is an algebra over an
-operad.
In an -operad, for any
, there is a space of
operations which are
-ary. There are
connected components, corresponding to the fact that none of the operations are anywhere near commutative, necessarily. But any connected component is supposed contractible.That means that, once you fix an order that elements to be composed in, i.e. a connected component, then the choice of
-ary operation is independent up to a contractible space. This is the sense in homotopy coherent life in which things are unique.
Now I want to convince you that a loop space is an -algebra. Let me describe a classical operad which acts on any loop space. This is called the little cubes operad, though I’ll only describe it in the case of little intervals. Here
is described as the space of linear imbeddings of
disjoint copies of
in itself. One can easily define a permutation law (switch around the imbedding) and a composition law (compose imbeddings). Moreover, one can see that this is an
operad. Once you choose an order for which the
copies of
are supposed to fit within
, then that determines everything; you can wiggle from one imbedding to another. So the little intervals operad is an
-operad
.
Note that the little intervals operad acts on the loop space
. Indeed, an element of
is just a loop on
. But a choice of imbeddings of
intervals in
gives one a way of composing
paths on
. So there are canonical maps
which correspond to the fact that the little intervals operad essentially parametrizes the different candidates for how you might compose paths in the loop space. One checks that this gives an operad action.
So, the big theorem is:
Theorem 2 Any connected space with an action of an
-operad is of the homotopy type of a loop space.
I will try to explain this in another post. Essentially, the strategy is to, given a space with an action of an operad, build a classifying space in the same way one can build a classifying space for a topological group. (A topological group
is the loop space of its classifying space
by the fiber sequence
where
is contractible.)
Leave a Reply