Algebra over an operad

Definition

Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a symmetric monoidal category, 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 of the symmetric group $\Sigma_n$ 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.