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 $n$


?



Are $E_k$ monoids higher categories?
This is closely related to the Baez–Dolan [stabilization hypothesis](https://ncatlab.org/nlab/show/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.


