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:
Variants and generalizations include non-symmetric (planar) operads that ignore input permutations and colored (multisorted) operads allowing