algebrënobject

Algebrënobject is a generalized algebraic structure defined within the framework of symmetric monoidal categories. It extends the classical notion of an algebra object by equipping an object with a family of multi-ary operations that generalize multiplication, enabling a broader range of algebraic behaviors. The term is used in contexts where operad theory provides a natural language for organizing these higher operations.

Formally, in a symmetric monoidal category C with tensor product ⊗ and unit object I, an algebrënobject consists of an object A together with a collection of morphisms μ_n: A ⊗ … ⊗ A → A for each n ≥ 2, and a unit morphism η: I → A. These maps are required to satisfy coherence relations encoded by an operad that acts on A. When all μ_n vanish for n ≠ 2 and μ_2 is associative, A is an ordinary associative algebra object. If a differential d: A → A is present with d^2 = 0 and a Leibniz-type rule holds, one obtains a differential graded algebra object. If the higher products μ_n satisfy the relations of an A∞-algebra, A is an A∞-algebra object. Different operads yield other familiar types, such as commutative, Poisson, or Gerstenhaber algebra objects.

Morphisms between algebrënobjects preserve the entire family of operations and the unit, i.e., a map f: A → B that commutes with all μ_n and η. Algebrënobjects thus provide a unified framework for studying a wide range of “higher” algebraic structures within category theory and homotopy theory.

See also: algebra object, operad, A∞-algebra, DG-algebra, monoidal category.