Contents

Idea

The little $k$-cubes operad is the operad or (∞,1)-operad whose $n$-ary operations are parameterized by rectilinear disjoint embeddings of $n$ $k$-dimensional cubes into another $k$-dimensional cube.

When regarded as a topological operad?, the topology on the space of all such embedding is such that a continuous path is given by continuously moving the images of these little cubes in the big cube around.

Therefore the algebras over the $E_k$ operad are ”$k$-fold monoidal” objects For instance k-tuply monoidal (n,r)-categories.

The limiting $E_{\mathrm{\infty }}$ operad is the resolution of the ordinary commutative monoid operad $\mathrm{Comm}$. Its algebras are homotopy commutative monoid objects such as $E_{\mathrm{\infty }}$-rings.

References

A discussion of $E_k$ in the context of (∞,1)-operads is in

• Jacob Lurie, $𝔼[k]$-Algebras (pdf)