operad algebra

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 $f:\; O\; \backslash to\; O\text{'}$ 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.{{harvnb|Francis|loc=Proposition 2.9.}}

• En-ring
• Homotopy Lie algebra

{{reflist}}

• {{cite web |first=John |last=Francis |url=http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf |title=Derived Algebraic Geometry Over $\backslash mathcal\{E\}\_n$-Rings }}
• {{cite arXiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|arxiv=q-alg/9702015}}

• {{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

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