nLab
Ek-Algebras

Idea

This are notes on

• Jacob Lurie, $𝔼[k]$-Algebras (pdf)

Using the definition of the notion (∞,1)-operad in terms of a vertical categorification of the notion of category of operators, the article discusses the ${𝔼}_k$- or little cubes operads and its En-algebras.

A major application in the second part of the article is the study of topological chiral homology.

Definitions and results

Grouplike monoid objects

Let $𝒳$ be an ∞-stack (∞,1)-topos and $X:$ 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 Assoc-monoid. Say that an $𝔼[k]$-algebra object in $𝒳$ is grouplike is 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.

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

The following result 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 reduces to the classical theorem by Peter 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.

Related literature

Lurie’s proof of the equivalence of $n+1$-connected objects with grouplike $E[k]$-objects is entirely at the level of (∞,1)-categories. One would hope that in addition there is a model for this equivalence at the level of model categories.

There is a model category structure on the category ${\mathrm{Top}}_{*}$ of pointed topological spaces, such that the cofibrant objects are $n$-connected CW-complexes, described in

• J. I. Extremiana Aldana, Luis Javier Hernández Paricio? and
  Maria Teresa Rivas Rodriguez, A closed model category for $(n-1)$-connected spaces , Proc. AMS, 124, Number 11, 1996 (pdf)

Stabilization hypothesis

A proof of the stabilization hypothesis for k-tuply monoidal n-categories is a byproduct of corollary 1.1.10, 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. The tensor product is defined in appendix B.7.

Deligne conjecture

Section 2.5 gives a proof of a generalization of the Deligne conjecture.

category: reference