In the theory of operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative.
(An operad that describes a multiplication that is both homotopy coherently associative and homotopy coherently commutative is called an E∞-operad.)
In the (usual) setting of operads with an action of the symmetric group on topological spaces, an operad A is said to be an A∞-operad if all of its spaces A(n) are Σn-equivariantly homotopy equivalent to the discrete spaces Σn (the symmetric group) with its multiplication action (where n ∈ N).
In other categories than topological spaces, the notions of homotopy and contractibility have to be replaced by suitable analogs, such as homology equivalences in the category of chain complexes.
The letter A in the terminology stands for "associative", and the infinity symbols says that associativity is required up to "all" higher homotopies.
More generally, there is a weaker notion of An-operad (n ∈ N), parametrizing multiplications that are associative only up to a certain level of homotopies.
-operad and the monoid π0(X) of its connected components is a group.
-operads in homotopy theory stems from this relationship between algebras over
Examples feature the Fukaya category of a symplectic manifold, when it can be defined (see also pseudoholomorphic curve).
This operad describes strictly associative multiplications.
A geometric example of an A∞-operad is given by the Stasheff polytopes or associahedra.
A less combinatorial example is the operad of little intervals: The space