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

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