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.)