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''.

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

Notes

References

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External links

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

Abstract algebra

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