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