A∞-operad

In mathematics, an A_∞-operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of associativity is loosened. In a simple associative operation, such as the multiplication of numbers, the order of operations does not matter: ${\displaystyle (a\times b)\times c=a\times (b\times c)}$. An algebraic structure governed by an A_∞-operad is one where this equality is not strict, but the two sides are connected by a homotopy (a continuous path or transformation). The operad itself provides the formal structure for these paths and for higher-level paths that ensure all possible ways of regrouping are compatible with each other.

More formally, an A_∞-operad is a parameter space for a multiplication map that is homotopy coherently associative. The "A" stands for "associative", and the infinity symbol "∞" indicates that the associativity holds up to an infinite hierarchy of higher homotopies. This concept is fundamental to the study of loop spaces and is a key tool in fields like symplectic geometry (through the Fukaya category). (An operad that describes a multiplication that is both homotopy coherently associative and homotopy coherently commutative is called an E_∞-operad.)

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.

An-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

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

The importance of ${\displaystyle A_{\infty }}$-operads in homotopy theory stems from this relationship between algebras over ${\displaystyle A_{\infty }}$-operads and loop spaces.

A∞-algebras

An algebra over the ${\displaystyle A_{\infty }}$-operad is called an ${\displaystyle A_{\infty }}$-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 ${\displaystyle A_{\infty }}$-operad is the associative operad a given by ${\displaystyle a(n)=\Sigma _{n}}$. This operad describes strictly associative multiplications. By definition, any other ${\displaystyle A_{\infty }}$-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 ${\displaystyle A(n)}$ consists of all embeddings of n disjoint intervals into the unit interval.

See also

• Homotopy associative algebra
• 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): 630–631. Retrieved 2008-01-17.
• J. Peter May (1972). The Geometry of Iterated Loop Spaces. Springer-Verlag. Archived from the original on 2015-07-07. Retrieved 2008-02-19.
• Martin Markl; Steve Shnider; Jim Stasheff (2002). Operads in Algebra, Topology and Physics. American Mathematical Society.
• 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.