---
tag: H5765CJ7VB
kind: definition
name: Algebra over an operad
msc: [18M60, 18M75, 55P48]
defines: ES9YATDWA3
uses: [5S7ER95XAC, 0515A6D2Y1, 0WCY42EV9V, 0N6R902FG1]
sources:
  - url: https://en.wikipedia.org/wiki/Operad_algebra
---

## Definition

Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a [[symmetric monoidal category|5S7ER95XAC]], such as sets or topological spaces with the cartesian product.
An *operad* $\mathcal{O}$ in $\mathcal{C}$ consists of objects $\mathcal{O}(n)$ for $n \ge 0$ and the following data.
There is a right [[action|0WCY42EV9V]] of the [[symmetric group|0515A6D2Y1]] $\Sigma_n$ on each $\mathcal{O}(n)$, and a unit $\eta \colon \mathbb{1} \to \mathcal{O}(1)$.
There are composition maps, which are associative, unital and equivariant:
$$\gamma \colon \mathcal{O}(k) \otimes \mathcal{O}(j_1) \otimes \cdots \otimes \mathcal{O}(j_k) \to \mathcal{O}(j_1 + \cdots + j_k).$$
An *algebra over $\mathcal{O}$*, or *$\mathcal{O}$-algebra*, is an object $A$ of $\mathcal{C}$ with maps
$$\theta_n \colon \mathcal{O}(n) \otimes A^{\otimes n} \to A \qquad (n \ge 0).$$
These maps must satisfy three conditions, written here with elements as for sets or spaces:

1. (associativity) $\theta_j(\gamma(c; d_1, \ldots, d_k); \mathbf{a}_1, \ldots, \mathbf{a}_k) = \theta_k(c; \theta_{j_1}(d_1; \mathbf{a}_1), \ldots, \theta_{j_k}(d_k; \mathbf{a}_k))$, where $j = j_1 + \cdots + j_k$ and $\mathbf{a}_i$ is a $j_i$-tuple of elements of $A$;
2. (unit) $\theta_1(1; a) = a$, where $1 = \eta(\ast) \in \mathcal{O}(1)$;
3. (equivariance) $\theta_n(c \cdot \sigma; a_1, \ldots, a_n) = \theta_n(c; a_{\sigma^{-1}(1)}, \ldots, a_{\sigma^{-1}(n)})$ for $\sigma \in \Sigma_n$.

In a general $\mathcal{C}$ the three clauses are the commutativity of the corresponding diagrams.

## Remarks

Equivalently, an $\mathcal{O}$-algebra structure on $A$ is a morphism of operads $\mathcal{O} \to \mathrm{End}_A$, where $\mathrm{End}_A(n)$ is the internal hom from $A^{\otimes n}$ to $A$, when $\mathcal{C}$ is closed.

Example: the commutative operad in sets has $\mathcal{O}(n) = \set{\ast}$ for every $n$.
An algebra is a set $A$ with one map $\theta_n \colon A^n \to A$ for each $n$.
Equivariance makes $\theta_2(a, b) = \theta_2(b, a)$, the associativity clause makes $\theta_2$ associative, and $\theta_0$ picks a unit.
So the algebras are exactly the commutative [[monoids|0N6R902FG1]].
Likewise the associative operad, with $\mathcal{O}(n) = \Sigma_n$, has the monoids as its algebras.

Non-example: take $A = \Z$ and try to make it an algebra over the commutative operad with $\theta_2(a, b) = a - b$.
Equivariance fails for the transposition, since $1 - 0 \ne 0 - 1$.