Operad algebra

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.

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 → O ′ {\displaystyle 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.^[1]

• En-ring
• Homotopy Lie algebra

• Francis, John. "Derived Algebraic Geometry Over E n {\displaystyle {\mathcal {E}}_{n}} -Rings" (PDF).
• Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.
• Vallette, Bruno Algebra + homotopy = operad, (2014).

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