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 ${n}$-fold loop space. An operad ${\mathcal{O}}$ 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 ${\mathcal{O}}$ is to give a space ${X}$ together with an interpretation of each of these “operations” as actual operations on ${X}$.

Motivation

The concrete motivation is as follows. Consider a pointed space ${(X, \ast)}$, the loop space ${\Omega X}$ 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 ${k}$ paths is equivalent to another way (i.e., differently parenthesized) of composing that collection of ${k}$ 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 ${\Omega X}$ 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 ${(\mathcal{C}, \otimes)}$, one usually does not want to require an equality of functors ${X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z}$; this is considered “evil.” Also, this does not generally happen in practice. Rather, one wants a canonical isomorphism between the functors

$\displaystyle X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z.$

But this alone is not enough. Given such an isomorphism, if you have ${n}$ 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 ${X \otimes (Y \otimes (Z \otimes W))}$ to ${((X \otimes Y) \otimes Z) \otimes W}$ 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 ${\Omega X}$: 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 ${(\infty, 1)}$-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 ${\mathcal{O}}$ is going to consist of sets ${\mathcal{O}(n)}$ of “${n}$-ary operations” (which you should think of as maps ${X^n \rightarrow X}$ for some set ${X}$) with the following data:

1. There is a composition law ${\mathcal{O}(k) \times \mathcal{O}(n_1) \times \dots \times \mathcal{O}(n_k) \rightarrow \mathcal{O}(n_1 + \dots + n_k)}$. This corresponds to the process of composing operations.
2. There is an action of the symmetric group ${S_n}$ on ${\mathcal{O}(n)}$.
3. There is a distinguished “identity element” in ${\mathcal{O}(1)}$ corresponding to the identity function.
4. There are standard associativity, unital, and symmetry constraints that one figures out from how composition of ordinary ${n}$-ary operations behaves.

The basic example is the operad of all ${n}$-ary operations (for ${n}$ varying!) on a set ${X}$, called the endomorphism operad. Given ${\mathcal{O}}$, we can say that ${\mathcal{O}}$ acts on ${X}$ if we are given a homomorphism of operads (which I won’t define) from ${\mathcal{O}}$ to the endomorphism operad of ${X}$. There is thus a category of sets with an ${\mathcal{O}}$-action, which is called the category of ${\mathcal{O}}$-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 ${\mathcal{O}}$, the connectivity of the spaces ${\mathcal{O}(n)}$ is a way of expressing homotopy coherence. For instance, let’s suppose the following. Each ${\mathcal{O}(n)}$ has ${\pi_0}$ equal to the symmetric group ${S_n}$, i.e. there are ${n!}$ components of ${\mathcal{O}(n)}$ permuted by the action of ${S_n}$. 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: ${\pi_0 \mathcal{O}}$ is the operad described below, the associative operad.)

Definition 1 Such an operad ${\mathcal{O}}$ is called an ${A_\infty}$-operad.

A simple example of an ${A_\infty}$-operad is the associative operad. In this case, each ${\mathcal{O}(n)}$ is just the discrete space ${S_n}$. Composition is defined in the natural way, via

$\displaystyle S_k \times S_{n_1} \times \dots \times S_{n_k} \rightarrow S_{n_1 + \dots + n_k}$

where the first term controls the order of composition. In other words, there are ${n!}$ ${n}$-ary operations, except that all are just distinct permutations of each other. So there is really one fundamental operation, in ${\mathcal{O}(2)}$, and everything else comes from there. One can check that an algebra over this operad is simply a topological monoid.

So an ${A_\infty}$-operad is supposed to be a “homotopy coherent thickening” of the associative operad. For instance, the loop space ${\Omega X}$ 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 ${A_\infty}$-operad.

In an ${A_\infty}$-operad, for any ${n}$, there is a space of ${n!}$ operations which are ${n}$-ary. There are ${n!}$ 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 ${n}$-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 ${A_\infty}$-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 ${\mathcal{O}(n)}$ is described as the space of linear imbeddings of ${n}$ disjoint copies of ${I = [0, 1]}$ 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 ${A_\infty}$ operad. Once you choose an order for which the ${n}$ copies of ${I}$ are supposed to fit within ${I}$, then that determines everything; you can wiggle from one imbedding to another. So the little intervals operad is an ${A_\infty}$-operad ${\mathcal{O}}$.

Note that the little intervals operad ${\mathcal{O}}$ acts on the loop space ${\Omega X}$. Indeed, an element of ${\Omega X}$ is just a loop on ${X}$. But a choice of imbeddings of ${n}$ intervals in ${I}$ gives one a way of composing ${n}$ paths on ${\Omega X}$. So there are canonical maps

$\displaystyle \mathcal{O}(n) \rightarrow \hom((\Omega X)^n, \Omega X)$

which correspond to the fact that the little intervals operad essentially parametrizes the different candidates for how you might compose ${n}$ 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 ${A_\infty}$-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 ${A_\infty}$ operad, build a classifying space in the same way one can build a classifying space for a topological group. (A topological group ${G}$ is the loop space of its classifying space ${BG}$ by the fiber sequence ${G \rightarrow EG \rightarrow BG}$ where ${EG }$ is contractible.)