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Ek-Algebras

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Idea

This are notes on

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\mathbb{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\mathcal{X} be an ∞-stack (∞,1)-topos and X:X : Assoc 𝒳\to \mathcal{X} be a monoid object in 𝒳\mathcal{X}. Say that XX is grouplike if the composite

Δ opAss𝒳 \Delta^{op} \to Ass \to \mathcal{X}

(see 1.1.13 of Commutative Algebra)

is a groupoid object in 𝒳\mathcal{X}.

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

Mon 𝔼[k] gp(𝒳)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]\mathbb{E}[k]-monoid objects.

Main result: kk-fold delooping, monoidalness and 𝔼[k]\mathbb{E}[k]-action

The following result makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that kk-fold delooping provides a correspondence betwen n-categories that have trivial r-morphisms for r<kr \lt k and k-tuply monoidal n-categories.

Theorem (k-tuply monoidal \infty-stacks)

Let k>0k \gt 0, let 𝒳\mathcal{X} be an ∞-stack (∞,1)-topos and let 𝒳 * k\mathcal{X}_*^{\geq k} denote the full subcategory of the category 𝒳 *\mathcal{X}_{*} of pointed objects, spanned by those pointed objects thar are k1k-1-connected (i.e. their first kk ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories

𝒳 * kMon 𝔼[k] 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=Top\mathcal{X} = Top, this reduces to the classical theorem by Peter May

Theorem (May recognition theorem)

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

Proof

This is EkAlg, theorem 1.3.16.

Lurie’s proof of the equivalence of n+1n+1-connected objects with grouplike E[k]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 Top *Top_* of pointed topological spaces, such that the cofibrant objects are nn-connected CW-complexes, described in

  • J. I. Extremiana Aldana, Luis Javier Hernández Paricio? and
    Maria Teresa Rivas Rodriguez, A closed model category for (n1)(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 \infty-operads such that

𝔼 k𝔼 k𝔼 k+k. \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

Revised on March 22, 2012 07:55:49 by Tim_Porter (95.147.237.122)