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}.


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…)