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.

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 ot 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 15, 2015 20:32:49 by Konrad Voelkel? (