Oalgebra

Oalgebra, or an O-algebra, is a mathematical structure defined relative to an operad O in a symmetric monoidal category. An operad O consists of a family of objects O(n) for n ≥ 0, equipped with actions of the symmetric group S_n and with composition maps that allow the plugging of outputs of n-ary operations into inputs of other operations. There is also a unit given by a map from the monoidal unit to O(1). An O-algebra A in the same category is an object together with a collection of structure maps θ_n: O(n) ⊗ A^{⊗ n} → A that realize the abstract n-ary operations of O as actual operations on A. These maps must be compatible with the operad composition and the symmetric group actions, encoding associativity, equivariance, and unit laws.

Examples help illustrate the idea. If O is the associative operad, then an O-algebra is an associative algebra. If O is the commutative operad, an O-algebra is a commutative algebra. The Lie operad yields Lie algebras, and more generally, algebras over various operads capture a wide range of algebraic structures. In topology, operads such as the little disks operad E_n give E_n-algebras, which model n-fold loop spaces up to homotopy.

Categories and constructions. The collection of all O-algebras in a given category forms a category Alg_O, with morphisms that preserve the O-structure. For suitable categories, there are notions of free O-algebras, adjunctions with forgetful functors, and model structures used in homotopical and homological algebra. O-algebras thus provide a unifying framework to study and compare different algebraic structures via the language of operads.