Operad

Definition

An operad $P$ in sets consists of the following data. For each integer $n \ge 0$ there is a set $P(n)$, whose elements are called operations of arity $n$. There is an element $1 \in P(1)$, the identity. For all $k \ge 1$ and $n_1, \ldots, n_k \ge 0$ there is a composition map $$\gamma \colon P(k) \times P(n_1) \times \cdots \times P(n_k) \to P(n_1 + \cdots + n_k).$$ For each $n$ the symmetric group $S_n$ has a right action on $P(n)$, written $f \mapsto f \cdot \sigma$. These data must satisfy:

1. Associativity: for $f \in P(k)$, $g_i \in P(n_i)$ and $h_j \in P(m_j)$ with $1 \le j \le N = n_1 + \cdots + n_k$, $$\gamma\big(\gamma(f; g_1, \ldots, g_k); h_1, \ldots, h_N\big) = \gamma\big(f; \gamma(g_1; h_1, \ldots, h_{n_1}), \ldots, \gamma(g_k; h_{N - n_k + 1}, \ldots, h_N)\big).$$
2. Unit: $\gamma(1; f) = f$ and $\gamma(f; 1, \ldots, 1) = f$ for all $f \in P(k)$.
3. Equivariance: the two identities below hold for $\sigma \in S_k$ and $\tau_i \in S_{n_i}$. $$\gamma(f \cdot \sigma; g_1, \ldots, g_k) = \gamma(f; g_{\sigma^{-1}(1)}, \ldots, g_{\sigma^{-1}(k)}) \cdot \sigma_{n_1, \ldots, n_k},$$ $$\gamma(f; g_1 \cdot \tau_1, \ldots, g_k \cdot \tau_k) = \gamma(f; g_1, \ldots, g_k) \cdot (\tau_1 \oplus \cdots \oplus \tau_k).$$ Here $\sigma_{n_1, \ldots, n_k} \in S_N$ cuts $\set{1, \ldots, N}$ into $k$ consecutive blocks of sizes $n_1, \ldots, n_k$. It moves the $i$-th block to the $\sigma(i)$-th place and keeps the order inside each block. The permutation $\tau_1 \oplus \cdots \oplus \tau_k \in S_N$ permutes the $i$-th block internally by $\tau_i$.