Algebra over an operad

In algebra, given an operad O (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.^[1]

See also

• En-ring
• Homotopy Lie algebra

Notes

1. & Francis, Proposition 2.9.

References

• John Francis, Derived Algebraic Geometry Over ${\mathcal {E}}_{n}$-Rings

External links

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