Little cubes operad

Example 12.1.7. For every \(k \geq 0\), there is a topological operad¹ called the little \(k\)-cubes operad, often denoted \(\Ee _k\). Its \(n\)-ary operations are the so-called ‘rectilinear’ embeddings \(\bigsqcup _{i=1}^n [0,1]^k \to [0,1]^k\) of a disjoint union of \(n\) copies of the \(k\)-dimensional cube into the \(k\)-dimensional cube. Here ‘rectilinear’ means that on each of the \(n\) components it is given by an affine function, i.e., maps of the form \[ [0,1]^k \to [0,1]^k, \quad (x_i)_{i=1}^k \mapsto (a_i x_i + b_i)_{i=1}^k, \qquad a_i > 0, \] which allow us to scale and shift the cube but not rotate or distort it in any other way. The composition in \(\Ee _k\) is given by composition of rectilinear embeddings. For \(k = 0\), this recovers the pointed operad \(\Ee _0\) described above.