Operad algebra explained

In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad replacing R.

Definitions

Given an operad O (say, a symmetric sequence in a symmetric monoidal ∞-category C), an algebra over an operad, or O-algebra for short, is, roughly, a left module over O with multiplications parametrized by O.

If O is a topological operad, then one can say an algebra over an operad is an O-monoid object in C. If C is symmetric monoidal, this recovers the usual definition.

Let C be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If

f: O \to O' is a map of operads and, moreover, if f is a homotopy equivalence, then the ∞-category of algebras over O in C is equivalent to the ∞-category of algebras over O in C.

See also

• En-ring
• Homotopy Lie algebra

References

• Web site: John . Francis . Derived Algebraic Geometry Over l{E} _{n} -Rings .
• Hinich. Vladimir. 1997-02-11. Homological algebra of homotopy algebras. q-alg/9702015.

External links

• • http://ncatlab.org/nlab/show/algebra+over+an+operad