symmetric monoidal (∞,1)-category of spectra
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.
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An algebra over an operad over $E_k$ is an Ek-algebra.
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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.
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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 $a_i \gt 0$.
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 $Rect(\square^k \times S, \square^k )$ denote the collection of all rectitlinear embeddings from $\square^k \times S$ into $\square^k$ . We will regard $Rect(\square^2\times S, \square^k )$ as a topological space (it can be identified with an open subset of $(\mathbf{R}^{2k} )^S )$.
The spaces $Rect(\square^k \times \{1, . . . , n\}, \square^k)$ constitute the $n$-ary operations of a topological operad, which we will denote by $tE_k$ and refer to as the little k-cubes operad.
This is Higher Algebra Definition 5.1.0.1.
We define a topological category $tE^\otimes_k$ as follows:
The objects of $t E^\otimes_k$ are the objects $[n] \in Fin_*$.
Given a pair of objects $[m], [n] \in tE^\otimes_k$ , a morphism from $[m]$ to $[n]$ in $t E^\otimes_k$ consists of the following data:
A morphism $\alpha : [m] \to [n]$ in $Fin_*$ .
For each $j \in [n]^\circ$ a rectilinear embedding $\square^k \times \alpha^{-1} \{j\} \to \square^k$.
For every pair of objects $[m], [n] \in tE^\otimes_k$ , we regard $Hom_{tE^\otimes_k} ([m], [n])$ as endowed with the topology induced by the presentation
Let $\mathcal{X}$ be an (∞,1)-sheaf (∞,1)-topos and $X : Assoc \to \mathcal{X}$ be a monoid object in $\mathcal{X}$. Say that $X$ is grouplike if the composite
(see 1.1.13 of Commutative Algebra)
is a groupoid object in $\mathcal{X}$.
Say an $\mathbb{E}[1]$-algebra object is grouplike if it is grouplike as an $Ass$-monoid. Say that an $\mathbb{E}[k]$-algebra object in $\mathcal{X}$ is grouplike if the restriction along $\mathbb{E}[1] \hookrightarrow \mathbb{E}[k]$ is. Write
for the (∞,1)-category of grouplike $\mathbb{E}[k]$-monoid objects.
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 $r \lt k$ and k-tuply monoidal n-categories.
Let $k \gt 0$, let $\mathcal{X}$ be an ∞-stack (∞,1)-topos and let $\mathcal{X}_*^{\geq k}$ denote the full subcategory of the category $\mathcal{X}_{*}$ 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
This is EKAlg, theorem 1.3.6..
Specifically for $\mathcal{X} = Top$, this refines to the classical theorem by (May).
Let $Y$ be a topological space equipped with an action of the little cubes operad $\mathcal{C}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.
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.
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
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.
Explicit models of $E_\infty$-operads include
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A standard textbook reference is chapter 4 of
John Francis’ work on $E_n$-actions on $(\infty,1)$-categories is in
This influenced the revised version of
and is extended to include a discussion of 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
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