operadic

Operadic is an adjective referring to operads, algebraic structures that encode operations with multiple inputs and a single output, together with rules for composing these operations and permuting inputs. The operadic framework generalizes algebraic theories such as associative, commutative, and Lie algebras and provides a language for organizing collections of operations that interact in specified ways. It is central in fields such as algebraic topology, category theory, and deformation theory.

An operad O consists of a sequence of objects O(n) for n ≥ 0, equipped with actions of the symmetric group S_n and with composition maps γ: O(k) × O(n1) × ... × O(nk) → O(n1 + ... + nk) satisfying associativity and unit axioms. An algebra over an operad O is a concrete object A together with structure maps O(n) × A^⊗n → A that are compatible with the operad's composition and symmetry. Operads thus parametrize the operations that a given algebraic structure can perform.

Prominent examples include the endomorphism operad End_V(n) = Hom(V^⊗n, V) for a vector space V, the associative operad that encodes associative algebras, the commutative operad for commutative algebras, and the Lie operad for Lie algebras. The little disks operad and related E_n-operads are central in topology, connecting operads to loop spaces and iterated homotopy. Variants include non-symmetric operads, colored (or multicategory) operads, and generalized frameworks such as PROPs.