### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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 ${E}_{\infty }$-rings.

## Definition

(…)

An algebra over an operad over ${E}_{k}$ is an Ek-algebra.

(…)

Remark

Many models for ${E}_{\infty }$-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 $\infty$-categories of E-∞ algebras.

(…)

### As $\infty$-operads

###### Definition

Fix an integer $k\ge 0$. We let ${\square }^{k}=\left(-1,1{\right)}^{k}$ 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 $f\left({x}_{1},...,{x}_{k}\right)=\left({a}_{1}{x}_{1}+{b}_{1},...,{a}_{k}{x}_{k}+{b}_{k}\right)$ 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 ${\square }^{k}×S\to {\square }^{k}$ is a rectilinear embedding if it is an open embedding whose restriction to each connected component of ${\square }^{k}×S$ is rectilinear.

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

The spaces $\mathrm{Rect}\left({\square }^{k}×\left\{1,...,n\right\},{\square }^{k}\right)$ 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 $\left[n\right]\in {\mathrm{Fin}}_{*}$.

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

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

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

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

${\mathrm{Hom}}_{{\mathrm{tE}}_{k}^{\otimes }}\left(\left[m\right],\left[n\right]\right)=\coprod _{f:\left[m\right]\to \left[n\right]}\prod _{1\le j\le n}\mathrm{Rect}\left(×{\alpha }^{-1}\left\{j\right\},{\square }^{k}\right)$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 $𝔼\left[1\right]$-algebra object is grouplike if it is grouplike as an $\mathrm{Ass}$-monoid. Say that an $𝔼\left[k\right]$-algebra object in $𝒳$ is grouplike if the restriction along $𝔼\left[1\right]↪𝔼\left[k\right]$ is. Write

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

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

### $k$-fold delooping, monoidalness and $𝔼\left[k\right]$-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 and k-tuply monoidal n-categories.

###### Theorem (k-tuply monoidal $\infty$-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

${𝒳}_{*}^{\ge k}\simeq {\mathrm{Mon}}_{𝔼\left[k\right]}^{\mathrm{gp}}\left(𝒳\right)\phantom{\rule{thinmathspace}{0ex}}.$\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.

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

${𝔼}_{k}\otimes {𝔼}_{k\prime }\simeq {𝔼}_{k+k\prime }\phantom{\rule{thinmathspace}{0ex}}.$\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.

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 ${E}_{\infty }$-operads include

(…)

## References

A standard textbook reference is chapter 4 of

John Francis’ work on ${E}_{n}$-actions on $\left(\infty ,1\right)$-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 ${E}_{k}$ in the context of (∞,1)-operads is in

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

For the relation to Poisson Operads see

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory : ${P}_{0}$-operad (wiki)

Revised on November 7, 2013 08:25:39 by Urs Schreiber (145.116.129.101)