Little cubes operad

Example

Fix $n \ge 1$ and let $I^n = (0, 1)^n$ be the open unit cube. A little $n$-cube is an embedding $I^n \to I^n$ of the form $(t_1, \ldots, t_n) \mapsto (a_1 + b_1 t_1, \ldots, a_n + b_n t_n)$ with all $b_i > 0$. The space $\mathcal{C}_n(k)$ is the topological space of $k$-tuples of little $n$-cubes with pairwise disjoint images. The symmetric group $\Sigma_k$ acts by permuting the $k$ cubes, and the unit is the identity cube in $\mathcal{C}_n(1)$. Composition inserts the configuration of $\mathcal{C}_n(j_i)$ into the $i$-th cube of an element of $\mathcal{C}_n(k)$, by composing embeddings. These data form the little $n$-cubes operad $\mathcal{C}_n$.

For a pointed space $X$, the $n$-fold loop space $\Omega^n X$ is an algebra over $\mathcal{C}_n$. View a point of $\Omega^n X$ as a map $\bar{I}^n \to X$ that sends the boundary to the base point. Given $c = (c_1, \ldots, c_k) \in \mathcal{C}_n(k)$ and loops $f_1, \ldots, f_k$, define $\theta_k(c; f_1, \ldots, f_k)$ to be $f_i \circ c_i^{-1}$ on the image of $c_i$ and the base point elsewhere. This map is continuous, and the algebra clauses hold because composing cubes corresponds to nesting the pictures.