Algebra over an operad

Summary

An operad lists abstract operations with several inputs and one output, together with rules for plugging operations into each other. An algebra over an operad is an object on which these abstract operations act as actual operations, compatibly with plugging, units and permutations of inputs. Monoids, commutative monoids, Lie algebras and loop spaces are all algebras over suitable operads. The notion lets one treat many algebraic structures, and their homotopy-coherent versions, in one framework.