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Operad algebra

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated June 30, 2026

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.

01Definitions

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.

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This article is adapted from the Wikipedia article “Operad algebra”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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