nLab 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


Given a natural number k0k \geq 0, write

k(1,1) k \square^k \;\coloneqq\; (-1, 1)^k

for the open cube of dimension kk (the kk-fold product topological space of the open interval with itself). We will say that a continuous map f: k kf \colon \square^k \to \square^k is a rectilinear embedding if, with respect to the canonical coordinate functions on (1,1)(-1,1) \,\subset\, \mathbb{R}, it is given by an affine function, hence by a formula of the form

f(x 1,,x k)=(a 1x 1+b 1,,a kx k+b k) f (x_1 , \cdots , x_k ) \;=\; (a_1 x_1 + b_1 , \cdots , a_k x_k + b_k )

for some real numbers a ia_i and b ib_i , with a i>0a_i \gt 0.

More generally, if SS is a finite set, call 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 in the above sense.

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, topologized as a subspace of the topological product space (R 2k) S\big(\mathbf{R}^{2k} \big)^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 k{}^{t} \mathbb{E}_k and refer to as the little k-cubes operad.

This is Definition in Higher Algebra.


We define a topological category t𝔼 k {}^t \mathbb{E}^\otimes_k as follows:

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

  • Given a pair of objects [m],[n] t𝔼 k [m], [n] \in {}^t \mathbb{E}^\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_{{}^t \mathbb{E}^\otimes_k} ([m], [n]) = \coprod_{f \colon [m]\to [n]} \prod_{1\le j\le n} Rect(\times \alpha^{-1} \{j\},\square^k) \,.
  • Composition of morphisms in t𝔼 k {}^t \mathbb{E}^\otimes_k is defined in the obvious way. We let 𝔼 k \mathbb{E}^\otimes_k denote the nerve of the topological category t𝔼 k {}^t \mathbb{E}^\otimes_k.

    Corollary T. implies that 𝔼 k \mathbb{E}^\otimes_k is an

\infty -category. There is an evident forgetful functor from t𝔼 k {}^t \mathbb{E}^\otimes_k to the (discrete) category Fin *Fin_* , which induces a functor 𝔼 k N(Fin *)\mathbb{E}^\otimes_k \to N(Fin_* ).

This is Higher Algebra Definition


Grouplike monoid objects

Let 𝒳\mathcal{X} be an (∞,1)-sheaf (∞,1)-topos

and X:Assoc𝒳X \colon 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 between n-categories that have trivial r-morphisms for r<kr \lt k and k-tuply monoidal n-categories.

Theorem (k-tuply monoidal ∞-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 that 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.

Relation to Fulton-MacPherson operad

(Salvatore 01, Prop. 4.9, summarized as Lambrechts-Volic 14, Prop. 5.6)

Cohomology: Formality

the little n-disk operad is formal

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 comprehensive reference for many known models is

Review includes

A standard textbook reference is chapter 4 of

The equivalence to the Fulton-MacPherson operad is due to

  • Paolo Salvatore, Configuration spaces with summable labels, Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 375-395.

Proof that the little n-disk operad is formal was sketched by Maxim Kontsevich and spelled out in

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

Last revised on March 4, 2023 at 21:27:18. See the history of this page for a list of all contributions to it.