Are $E_k$ monoids higher categories?

Question

The May Recognition Theorem establishes an equivalence between the $\infty$-categories

• The $\infty$-category of grouplike $E_n$ monoids
• The $\infty$-category of pointed $(n-1)$-connected spaces

There is also an equivalence between the $\infty$-categories

• The $\infty$-category of $E_1$ monoids
• The $\infty$-category of pointed $(\infty,1)$-categories whose core is a connected space

Is there analogous generalization to higher dimension? E.g. something like an equivalence between

• The $\infty$-category of $E_n$ monoids
• The $\infty$-category of pointed $(\infty,n)$-categories that are sufficiently trivial in dimensions below $n$

?

Answer 1

This is closely related to the Baez–Dolan stabilization hypothesis.

There are numerous proofs of this statement. One line of reasoning is to establish a general 1-category statement first: given a symmetric monoidal presentable (∞,1)-category $C$, the (∞,1)-category of $C$-enriched categories with one object is equivalent to the (∞,1)-category of ∞-monoids in $C$, i.e., algebras over the operad $\def\E{{\rm E}} \E_1$ in $C$.

Iterating this result $n$ times, we obtain that the (∞,1)-category of categories enriched in the (∞,1)-category of categories enriched in … (repeat $n$ times) … in $C$, with a single $k$-morphism for all $0≤k<n$ is equivalent to the (∞,1)-category of $\E_1$-algebras in $\E_1$-algebras in … (repeat $n$ times) … in $\E_1$-algebras in $C$.

By Dunn's additivity theorem, the latter (∞,1)-category is equivalent to $\E_n$-algebras in $C$.

Depending on your preferences, this sketch can be formalized using model categories, quasicategories, etc., see the nLab article cited above for some references.