# Operad algebra

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In algebra, an '''operad algebra''' is an "algebra" over an [operad](/source/operad). It is a generalization of an [associative algebra](/source/associative_algebra) over a commutative ring ''R'', with an operad replacing ''R''.

== Definitions ==
Given an operad ''O'' (say, a [symmetric sequence](/source/symmetric_sequence) in a [symmetric monoidal](/source/symmetric_monoidal) [∞-category](/source/%E2%88%9E-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](/source/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 <math>f: O \to O'</math> 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''.<ref>{{harvnb|Francis|loc=Proposition 2.9.}}</ref>

== See also ==
*[En-ring](/source/En-ring)
*[Homotopy Lie algebra](/source/Homotopy_Lie_algebra)

== Notes ==
{{reflist}}

== References ==
*{{cite web |first=John |last=Francis |url=http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf |title=Derived Algebraic Geometry Over <math>\mathcal{E}_n</math>-Rings }}
*{{cite arXiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|arxiv=q-alg/9702015 }}
*Vallette, Bruno [https://arxiv.org/pdf/1202.3245 Algebra + homotopy = operad], (2014).

== External links ==
*{{citation |title=operad |url=http://ncatlab.org/nlab/show/operad |website=ncatlab.org}}
*http://ncatlab.org/nlab/show/algebra+over+an+operad

Category:Abstract algebra

{{abstract-algebra-stub}}

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