Operad

A collection of spaces (or sets, vector spaces, etc.) O(n) for each natural number n (arity) together with composition maps and unit satisfying associativity, equivariance (when symmetric), and unit laws; encodes types of algebras by specifying abstract n-ary operations and how they compose.

An algebraic structure that encodes collections of operations with multiple inputs and one output, together with composition maps and (in the symmetric case) actions of symmetric groups that express permuting inputs.