In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad ${\displaystyle O}$, one defines an algebra over ${\displaystyle O}$ to be a set together with concrete operations on this set which behave just like the abstract operations of ${\displaystyle O}$. For instance, there is a Lie operad ${\displaystyle L}$ such that the algebras over ${\displaystyle L}$ are precisely the Lie algebras; in a sense ${\displaystyle L}$ abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.

Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972.

Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:

The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer).

Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology in the work of Maxim Kontsevich and Thomas Willwacher.

Suppose ${\displaystyle X}$ is a set and for ${\displaystyle n\in \mathbb {N} }$ we define

the set of all functions from the cartesian product of ${\displaystyle n}$ copies of ${\displaystyle X}$ to ${\displaystyle X}$.

We can compose these functions: given ${\displaystyle f\in P(n)}$, ${\displaystyle f_{1}\in P(k_{1}),\ldots ,f_{n}\in P(k_{n})}$, the function