In the theory of operads in algebra and algebraic topology, an **E**_{∞}-operad is a parameter space for a multiplication map that is associative and commutative "up to all higher homotopies". (An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an A_{∞}-operad.)

For the definition, it is necessary to work in the category of operads with an action of the symmetric group. An operad *A* is said to be an E_{∞}-operad if all of its spaces *E*(*n*) are contractible; some authors also require the action of the symmetric group *S*_{n} on *E*(*n*) to be free. In other categories than topological spaces, the notion of *contractibility* has to be replaced by suitable analogs, such as acyclicity in the category of chain complexes.

The letter *E* in the terminology stands for "everything" (meaning associative and commutative), and the infinity symbols says that commutativity is required up to "all" higher homotopies. More generally, there is a weaker notion of *E*_{n}-operad (*n* ∈ **N**), parametrizing multiplications that are commutative only up to a certain level of homotopies. In particular,

*E*_{1}-spaces are *A*_{∞}-spaces;
*E*_{2}-spaces are homotopy commutative *A*_{∞}-spaces.
The importance of *E*_{n}- and *E*_{∞}-operads in topology stems from the fact that iterated loop spaces, that is, spaces of continuous maps from an *n*-dimensional sphere to another space *X* starting and ending at a fixed base point, constitute algebras over an *E*_{n}-operad. (One says they are *E*_{n}-spaces.) Conversely, any connected *E*_{n}-space *X* is an *n*-fold loop space on some other space (called *B*^{n}X, the *n*-fold classifying space of X).

The most obvious, if not particularly useful, example of an *E*_{∞}-operad is the *commutative operad* *c* given by *c*(*n*) = *, a point, for all *n*. Note that according to some authors, this is not really an *E*_{∞}-operad because the *S*_{n}-action is not free. This operad describes strictly associative and commutative multiplications. By definition, any other *E*_{∞}-operad has a map to *c* which is a homotopy equivalence.

The operad of **little ***n*-cubes or **little ***n*-disks is an example of an *E*_{n}-operad that acts naturally on *n*-fold loop spaces.