nLab
little cubes operad

Idea

The little $k$-disk operad or little $k$-cubes operad (to distinguish from the framed little n-disk operad) is the topological operad/(∞,1)-operad $E_k$ 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-∞ operad is a resolution of the ordinary commutative monoid operad Comm. Its algebras are homotopy commutative monoid objects such as $E_{\mathrm{\infty }}$-rings.

Definition

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An algebra over an operad over $E_k$ is an Ek-algebra.

Presentation by enriched operads

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Remark

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

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As $\mathrm{\infty }$-operads

Definition

Fix an integer $k\ge 0$. We let ${\square }^k=(-1,1{)}^k$ denote an open cube of dimension $k$. We will say that a map $f:{\square }^k\to {\square }^k$ is a rectilinear embedding if it is given by the formula $f(x_1,...,x_k)=(a_1{x}_1+b_1,...,a_k{x}_k+b_k)$ for some real constants $a_i$ and $b_i$ , with $a_i>0$.

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

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

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

This is Higher Algebra Definition 5.1.0.1.

Definition

We define a topological category ${\mathrm{tE}}_k^{\otimes }$ as follows:

• The objects of $t{E}_k^{\otimes }$ are the objects $[n]\in {\mathrm{Fin}}_{*}$.

• Given a pair of objects $[m],[n]\in {\mathrm{tE}}_k^{\otimes }$ , a morphism from $[m]$ to $[n]$ in $t{E}_k^{\otimes }$ consists of the following data:

  • A morphism $\alpha :[m]\to [n]$ in ${\mathrm{Fin}}_{*}$ .

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

• For every pair of objects $[m],[n]\in {\mathrm{tE}}_k^{\otimes }$ , we regard ${\mathrm{Hom}}_{{\mathrm{tE}}_k^{\otimes }}([m],[n])$ as endowed with the topology induced by the presentation

$${\mathrm{Hom}}_{{\mathrm{tE}}_k^{\otimes }}([m],[n])=\coprod _{f:[m]\to [n]}\prod _{1\le j\le n}\mathrm{Rect}(\times {\alpha }^{-1}\{j\},{\square }^k)$$Hom_{tE^\otimes_k} ([m], [n]) = \coprod_{f : [m]\to [n]} \prod_{1\le j\le n}Rect(\times \alpha^{-1} \{j\},\square^k)$$. * Composition of morphisms in $tE^\otimes_k$ is defined in the obvious way. We let $E^\otimes_k$ denote the nerve of the topological category $tE^\otimes_k$ . Corollary T.1.1.5.12 implies that $E^\otimes_k is an $\infty$-category. There is an evident forgetful functor from $tE^\otimes_k$ to the (discrete) category $Fin_*$ , which induces a functor $E^\otimes_k \to N(Fin_* )$. This is [[Higher Algebra]] Definition 5.1.0.2.

Properties

Grouplike monoid objects

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

$${\Delta }^{\mathrm{op}}\to \mathrm{Ass}\to 𝒳$$\Delta^{op} \to Ass \to \mathcal{X}

(see 1.1.13 of Commutative Algebra)

is a groupoid object in $𝒳$.

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

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

for the (∞,1)-category of grouplike $𝔼[k]$-monoid objects.

$k$-fold delooping, monoidalness and $𝔼[k]$-action

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

Theorem (k-tuply monoidal $\mathrm{\infty }$-stacks)

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

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

Proof

This is EKAlg, theorem 1.3.6..

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

Theorem (May recognition theorem)

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

Proof

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 $\mathrm{\infty }$-operads such that

$${𝔼}_k\otimes {𝔼}_{k\prime }\simeq {𝔼}_{k+k\prime }.$$\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 $E_k$-operad in a category of chain complexes, its homology is the Poisson operad? $P_k$.

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

Examples

Explicit models of $E_{\mathrm{\infty }}$-operads include

• Barratt-Eccles operad

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Related concepts

• A-∞ operad

• E-∞ operad

• L-∞ operad

References

A standard textbook reference is chapter 4 of

• Peter May, The geometry of iterated loop spaces (pdf)

John Francis’ work on $E_n$-actions on $(\mathrm{\infty },1)$-categories is in

• John Francis PhD thesis (web)

This influenced the revised version of

• Jacob Lurie, Commutative Algebra (arXiv)

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

• David Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld Centers in Derived Geometry (arXiv)

  (see also geometric ∞-function theory)

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

• Jacob Lurie, $𝔼[k]$-Algebras (pdf)

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

• Dev Sinha, The homology of the little disks operad (arXiv:math/0610236)

For the relation to Poisson Operads see

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory : $P_0$-operad (wiki)