---
tag: FWT0003QS1
kind: example
name: Little cubes operad
msc: [18M75, 55P48, 55P35]
illustrates: [ES9YATDWA3]
uses: [ES9YATDWA3, 2JFF5F3EGF, 0515A6D2Y1, T78YG90VVT, 0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/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|0CQHFPGJ45]] of $k$-tuples of little $n$-cubes with pairwise disjoint images.
The [[symmetric group|0515A6D2Y1]] $\Sigma_k$ 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|2JFF5F3EGF]] $\Omega^n X$ is an [[algebra|ES9YATDWA3]] 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.

## Remarks

The space $\mathcal{C}_n(k)$ is homotopy equivalent to the space of ordered configurations of $k$ distinct points in $\R^n$, by taking the centres of the cubes.
In particular $\mathcal{C}_n(2)$ is homotopy equivalent to the sphere $S^{n-1}$.
For $n = 1$ the operations are associative up to homotopy, but not commutative, since $\mathcal{C}_1(2)$ has two contractible components, one for each order.
For $n \ge 2$ the sphere $S^{n-1}$ is connected, which is the reason why $\pi_2$ and higher homotopy groups are abelian.
May's recognition principle says that a group-like $\mathcal{C}_n$-algebra is weakly equivalent to an $n$-fold loop space.
Letting $n \to \infty$ gives an operad $\mathcal{C}_\infty$, which is an [[E-infinity operad|T78YG90VVT]].

## Explanations

### L1 human

Picture a big square with several small squares drawn inside it, none overlapping.
This picture is a rule for combining several maps defined on squares into one: shrink each map into its small square and fill the rest with a constant.
Such pictures can be nested, and that nesting is the operad structure.
In dimension one the small intervals sit in a line and their order matters, which gives associative but not commutative products.
In dimension two or more the small squares can move around each other, which makes the products commutative up to homotopy.