A_∞-operad

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

Contents 1 Definition 2 An-operads 3 A∞-operads and single loop spaces 4 A∞-algebras 5 Examples 6 See also 7 References

Definition

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.

A_n-operads

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,

• A₁-spaces are pointed spaces;
• A₂-spaces are H-spaces with no associativity conditions; and
• A₃-spaces are homotopy associative H-spaces.

A_∞-operads and single loop spaces

The importance of A_∞-operads in topology stems from the fact that loop spaces, that is, spaces of continuous maps from the unit circle to another space X starting and ending at a fixed base point, constitute algebras over an A_∞-operad. (One says they are A_∞-spaces.) Conversely, any connected A_∞-space X is, up to homotopy, the loop space of some other space (called BX, the classifying space of X). For disconnected spaces A_∞-spaces X, the group completion of X is always a loop space, but X itself might not be one.

A_∞-algebras

An algebra over the A_∞ operad is called an A_∞-algebra. Examples feature the Fukaya category of a symplectic manifold, when it can be defined (see also pseudoholomorphic curve).

Examples

The most obvious, if not particularly useful, example of an A_∞-operad is the associative operad a given by a(n) = Σ_n. This operad describes strictly associative multiplications. By definition, any other A_∞-operad has a map to a which is a homotopy equivalence.

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 A(n) consists of all embeddings of n disjoint intervals into the unit interval.

See also

• operad
• E-infinity operad
• loop space

References

• Stasheff, Jim (June/July 2004). "What Is...an Operad?" (PDF). Notices of the American Mathematical Society 51 (6): pp.630–631. http://www.ams.org/notices/200406/what-is.pdf. Retrieved 2008-01-17. 

• J. P. May (1972). The Geometry of Iterated Loop Spaces. Springer-Verlag. http://www.math.uchicago.edu/~may/BOOKSMaster.html. 

• Martin Markl, Steve Shnider, Jim Stasheff (2002). Operads in Algebra, Topology and Physics. American Mathematical Society. http://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-96. 

• Stasheff, James (1963). "Homotopy associativity of H-spaces. I, II.". Transactions of the American Mathematical Society 108 (2): 275–292; 293–312. doi:10.2307/1993608. JSTOR 1993608.