An operad axiomatises many-in, one-out composition. It consists of a family of arity- operation objects carrying a symmetric-group action, together with equivariant composition maps that are associative and unital. Equivalently, is a monoid for the composition product on symmetric sequences. A coloured operad, equivalently a symmetric multicategory, types every input and output, so an operation reads . Its algebras interpret the abstract operations as concrete multilinear maps. The properad/polycategory generalisation allows many outputs at once. An operad properly extends a category, whose composition recovers the binary case of chaining at . Two prototypes are the little-discs operad and the endomorphism operad of an object in a symmetric monoidal category, whose algebras are precisely the algebraic structures borne by .
References
J. Peter May, “The Geometry of Iterated Loop Spaces” (1972)
Tom Leinster, “Higher Operads, Higher Categories” (2004)