Operad

An operad is a mathematical framework that encodes how to compose operations with multiple inputs into a single output. In its common form, an operad P consists of a sequence of objects P(n) for n ≥ 0, each equipped with a right action of the symmetric group S_n, together with composition maps gamma: P(k) × P(n1) × ... × P(nk) → P(n1+...+nk) and a distinguished unit element in P(1). These data satisfy associativity and unit axioms, and the symmetry action is compatible with the compositions.

A P-algebra on an object X in the same category provides concrete realizations of the abstract operations: X carries maps P(n) × X^n → X that respect the symmetry and composition. End(X) is a standard example, with End(X)(n) = Hom(X^n, X); algebras over End(X) are just ordinary algebraic structures on X. Other familiar examples include the Com operad, where Com(n) is a singleton with a trivial S_n-action, whose algebras are commutative associative algebras, and the Lie operad, whose algebras are Lie algebras. In topology, operads such as the little disks operad encode higher homotopy structures; algebras over E_n operads model n-fold loop space structures.

Variants and generalizations include non-symmetric (planar) operads that ignore input permutations and colored (multisorted) operads allowing multiple types of inputs and outputs. Categorical perspectives view operads as monoids in the category of symmetric sequences, with algebras over an operad corresponding to algebras for an associated monad. Originating in algebraic topology in the 1970s, operads have since influenced algebra, category theory, and mathematical physics.