little cubes operad



The little kk-disk operad or little kk-cubes operad (to distinguish from the framed little n-disk operad) is the topological operad/(∞,1)-operad E kE_k whose nn-ary operations are parameterized by rectilinear disjoint embeddings of nn kk-dimensional cubes into another kk-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 kE_k operad are “kk-fold monoidal” objects For instance k-tuply monoidal (n,r)-categories.

The limiting E-∞ operad is a resolution of the ordinary commutative monoid operad Comm. Its algebras are homotopy commutative monoid objects such as E E_\infty-rings.



An algebra over an operad over E kE_k is an Ek-algebra.

Presentation by enriched operads



Many models for E E_\infty-operads in the literature are not in fact cofibrant in the model structure on operads, but are Σ\Sigma-cofibrant. By the therem at model structure on algebras over an operad, this is sufficient for their categories of algebras to present the correct \infty-categories of E-∞ algebras.


As \infty-operads


Fix an integer k0k \ge 0. We let k=(1,1) k\square^k = ( -1, 1)^k denote an open cube of dimension kk. We will say that a map f: k kf : \square^k \to \square^k is a rectilinear embedding if it is given by the formula f(x 1,...,x k)=(a 1x 1+b 1,...,a kx k+b k)f (x_1 , . . . , x_k ) = (a_1 x_1 + b_1 , . . . , a_k x_k + b_k ) for some real constants a ia_i and b ib_i , with a i>0a_i \gt 0.

More generally, if SS is a finite set, then we will say that a map k×S k\square^k \times S \to \square^k is a rectilinear embedding if it is an open embedding whose restriction to each connected component of k×S\square^k\times S is rectilinear.

Let Rect( k×S, k)Rect(\square^k \times S, \square^k ) denote the collection of all rectitlinear embeddings from k×S\square^k \times S into k\square^k . We will regard Rect( 2×S, k)Rect(\square^2\times S, \square^k ) as a topological space (it can be identified with an open subset of (R 2k) S)(\mathbf{R}^{2k} )^S ).

The spaces Rect( k×{1,...,n}, k)Rect(\square^k \times \{1, . . . , n\}, \square^k) constitute the nn-ary operations of a topological operad, which we will denote by tE ktE_k and refer to as the little k-cubes operad.

This is Higher Algebra Definition


We define a topological category tE k tE^\otimes_k as follows:

  • The objects of tE k t E^\otimes_k are the objects [n]Fin *[n] \in Fin_*.

  • Given a pair of objects [m],[n]tE k [m], [n] \in tE^\otimes_k , a morphism from [m][m] to [n][n] in tE k t E^\otimes_k consists of the following data:

    • A morphism α:[m][n]\alpha : [m] \to [n] in Fin *Fin_* .

    • For each j[n] j \in [n]^\circ a rectilinear embedding k×α 1{j} k\square^k \times \alpha^{-1} \{j\} \to \square^k.

  • For every pair of objects [m],[n]tE k [m], [n] \in tE^\otimes_k , we regard Hom tE k ([m],[n])Hom_{tE^\otimes_k} ([m], [n]) as endowed with the topology induced by the presentation

Hom tE k ([m],[n])= f:[m][n] 1jnRect(×α 1{j}, k) Hom_{tE^\otimes_k} ([m], [n]) = \coprod_{f : [m]\to [n]} \prod_{1\le j\le n}Rect(\times \alpha^{-1} \{j\},\square^k)


Grouplike monoid objects

Let 𝒳\mathcal{X} be an (∞,1)-sheaf (∞,1)-topos and X:Assoc𝒳X : Assoc \to \mathcal{X} be a monoid object in 𝒳\mathcal{X}. Say that XX is grouplike if the composite

Δ opAss𝒳 \Delta^{op} \to Ass \to \mathcal{X}

(see 1.1.13 of Commutative Algebra)

is a groupoid object in 𝒳\mathcal{X}.

Say an 𝔼[1]\mathbb{E}[1]-algebra object is grouplike if it is grouplike as an AssAss-monoid. Say that an 𝔼[k]\mathbb{E}[k]-algebra object in 𝒳\mathcal{X} is grouplike if the restriction along 𝔼[1]𝔼[k]\mathbb{E}[1] \hookrightarrow \mathbb{E}[k] is. Write

Mon 𝔼[k] gp(𝒳)Mon 𝔼[k](𝒳) Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \subset Mon_{\mathbb{E}[k]}(\mathcal{X})

for the (∞,1)-category of grouplike 𝔼[k]\mathbb{E}[k]-monoid objects.

kk-fold delooping, monoidalness and 𝔼[k]\mathbb{E}[k]-action

The following result of (Lurie) makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that kk-fold delooping provides a correspondence betwen n-categories that have trivial r-morphisms for r<kr \lt k and k-tuply monoidal n-categories.

Theorem (k-tuply monoidal \infty-stacks)

Let k>0k \gt 0, let 𝒳\mathcal{X} be an ∞-stack (∞,1)-topos and let 𝒳 * k\mathcal{X}_*^{\geq k} denote the full subcategory of the category 𝒳 *\mathcal{X}_{*} of pointed objects, spanned by those pointed objects thar are k1k-1-connected (i.e. their first kk ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories

𝒳 * kMon 𝔼[k] gp(𝒳). \mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,.

This is EKAlg, theorem 1.3.6..

Specifically for 𝒳=Top\mathcal{X} = Top, this refines to the classical theorem by (May).

Theorem (May recognition theorem)

Let YY be a topological space equipped with an action of the little cubes operad 𝒞 k\mathcal{C}_k and suppose that YY is grouplike. Then YY is homotopy equivalent to a kk-fold loop space Ω kX\Omega^k X for some pointed topological space XX.


This is EkAlg, theorem 1.3.16.

Proofs independent of higher order categories can be extracted from the literature. See this MO answer by Tyler Lawson for details.

Stabilization hypothesis

A proof of the stabilization hypothesis for k-tuply monoidal n-categories is a byproduct of corollary 1.1.10 of (Lurie), stated as example 1.2.3.

Additivity theorem

It has been long conjectured that it should be true that when suitably defined, there is a tensor product of \infty-operads such that

𝔼 k𝔼 k𝔼 k+k. \mathbb{E}_k \otimes \mathbb{E}_{k'} \simeq \mathbb{E}_{k + k'} \,.

This is discussed and realized in section 1.2. of (Lurie). The tensor product is defined in appendix B.7.

Homology: Poisson operads

For an E kE_k-operad in a category of chain complexes, its homology is the Poisson operad? P kP_{k}.

See for instance (Costello) and see at Poisson n-algebra.


Explicit models of E E_\infty-operads include



A standard textbook reference is chapter 4 of

John Francis’ work on E nE_n-actions on (,1)(\infty,1)-categories is in

This influenced the revised version of

and is extended to include a discussion of traces and centers in

A detailed discussion of E kE_k in the context of (∞,1)-operads is in

An elementary computation of the homology of the little nn-disk operad in terms of solar system calculus is in

For the relation to Poisson Operads see

Revised on March 30, 2017 14:48:04 by Arun Debray? (