A∞-algebra

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 Σ_n-equivariantly homotopy equivalent to the discrete spaces Σ_n (the symmetric group) with its multiplication action (where n ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad A is A_∞if all of its spaces A(n) are contractible. 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 A_n-operad (n ∈ N), parametrizing multiplications that are associative only up to a certain level of homotopies. In particular,

A space X is the loop space of some other space, denoted by BX, if and only if X is an algebra over an A_∞-operad and the monoid π₀(X) of its connected components is a group. An algebra over an A_∞-operad is referred to as an A_∞-space. There are three consequences of this characterization of loop spaces. First, a loop space is an A_∞-space. Second, a connected A_∞-space X is a loop space. Third, the group completion of a possibly disconnected A_∞-space is a loop space.

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