symmetric monoidal (∞,1)-category of spectra
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 $\mathbb{E}_k$- 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.
Let $\mathcal{X}$ be an ∞-stack (∞,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 Assoc-monoid. Say that an $\mathbb{E}[k]$-algebra object in $\mathcal{X}$ is grouplike is 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 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 reduces to the classical theorem by Peter May
Let $Y$ be a topological space equipped with an action of the little cubes operad $\mathcal{C}_k$ 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 $X$.
This is EkAlg, theorem 1.3.16.
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 $Top_*$ of pointed topological spaces, such that the cofibrant objects are $n$-connected CW-complexes, described in
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
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. The tensor product is defined in appendix B.7.
Section 2.5 gives a proof of a generalization of the Deligne conjecture.