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 $E_\infty$-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 $E_\infty$-operads in the literature are not in fact cofibrant in the model structure on operads, but are $\Sigma$-cofibrant. By the theorem 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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As $\infty$-operads

Definition

Given a natural number $k \geq 0$, write

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

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

$$f (x_1 , \cdots , x_k ) \;=\; (a_1 x_1 + b_1 , \cdots , a_k x_k + b_k )$$

for some real numbers $a_i$ and $b_i$ , with $a_i \gt 0$.

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

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, topologized as a subspace of the topological product space $\big(\mathbf{R}^{2k} \big)^S$.

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

This is Definition 5.1.0.1 in Higher Algebra.

Definition

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

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

• Given a pair of objects $[m], [n] \in {}^t \mathbb{E}^\otimes_k$ , 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

  $$Hom_{{}^t \mathbb{E}^\otimes_k} ([m], [n]) = \coprod_{f \colon [m]\to [n]} \prod_{1\le j\le n} Rect(\times \alpha^{-1} \{j\},\square^k) \,.$$

• Composition of morphisms in ${}^t \mathbb{E}^\otimes_k$ is defined in the obvious way. We let $\mathbb{E}^\otimes_k$ denote the nerve of the topological category ${}^t \mathbb{E}^\otimes_k$.

  Corollary T.1.1.5.12 implies that $\mathbb{E}^\otimes_k$ is an

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

This is Higher Algebra Definition 5.1.0.2.

Properties

Grouplike monoid objects

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

and $X \colon Assoc \to \mathcal{X}$ be a monoid object in $\mathcal{X}$. Say that $X$ is grouplike if the composite

$$\Delta^{op} \to Assoc \to \mathcal{X}$$

(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 $Assoc$-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

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

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

$k$-fold delooping, monoidalness and $\mathbb{E}[k]$-action

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

Theorem (k-tuply monoidal ∞-stacks)

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 that are $k-1$-connected (i.e. their first $k$ ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories

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

Proof

This is EKAlg, theorem 1.3.6..

Specifically for $\mathcal{X} = Top$, 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 $\mathcal{C}_k$ 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 $X$.

Proof

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

$$\mathbb{E}_k \otimes \mathbb{E}_{k'} \simeq \mathbb{E}_{k + k'} \,.$$

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

Proposition

The Fulton-MacPherson operad is weakly equivalent in the model structure on operads with respect to the classical model structure on topological spaces, to the little n-disk 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 $E_k$-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

• Barratt-Eccles operad

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Related concepts

• configuration space of points, Fulton-MacPherson operad

• factorization algebra

• A-∞ operad

• E-∞ operad

• L-∞ operad

References

A comprehensive reference for many known models is

• André Beuckelmann?, Ieke Moerdijk, A small catalogue of $E_n$-operads, arXiv:2212.03842v1.

Review includes

• Benoit Fresse, Little discs operads, graph complexes and Grothendieck–Teichmüller groups, in Haynes Miller (ed.) Handbook of Homotopy Theory (arXiv:1811.12536)

A standard textbook reference is chapter 4 of

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

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

• Pascal Lambrechts, Ismar Volić, section 5 of Formality of the little N-disks operad, Memoirs of the American Mathematical Society ; no. 1079, 2014 (doi:10.1090/memo/1079)

John Francis‘ work on $E_n$-actions on $(\infty,1)$-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 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

• Jacob Lurie, $\mathbb{E}[k]$-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