E∞-operad
# E∞-operad

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 $A_∞-operad.)$

## Definition

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.

## $E_n-operads$ and n-fold loop spaces

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 $B^nX,$ the n-fold classifying space of X).

## Examples

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.

