**The classical setting.**

Given a monoid $A$, there's a category $\mathbf{B}A$, called the **delooping** of $A$, having a single object $\star$ and satisfying $\mathrm{Hom}_{\mathbf{B}A}(\star,\star)\overset{\mathrm{def}}{=}A$, with composition given by multiplication and the sole identity $\mathrm{id}_{\star}$ given by $1_A$.

This construction is characterised by the following property: for any other category $\mathcal{C}$, we have a bijection of sets $$ \left\{ \begin{gathered} \text{functors}\\ \mathbf{B}A\to\mathcal{C} \end{gathered} \right\} \cong \left\{ \begin{aligned} &\text{pairs $(X,\phi)$ with}\\ &\,\,\,\,\,\,\,\text{- $X$ an object of $\mathcal{C}$;}\\ &\,\,\,\,\,\,\,\text{- $\phi$ a morphism of monoids}\\ &\text{from $A$ to $\left(\mathrm{Hom}_{\mathcal{C}}(X,X),\circ,\mathrm{id}_{X}\right)$.} \end{aligned} \right\}. $$ For example:

- A functor $\mathbf{B}\mathbb{N}\to\mathcal{C}$ is the same as an
*endomorphism*$A\to A$ of $\mathcal{C}$; - A functor $\mathbf{B}\mathbb{Z}\to\mathcal{C}$ is the same as an
*automorphism*$A\to A$ of $\mathcal{C}$; - A functor $\mathbf{B}\mathbb{Z}/2\to\mathcal{C}$ is the same as an
*involution*$A\to A$ of $\mathcal{C}$; - A functor $\mathbf{B}\mathbb{B}\to\mathcal{C}$ is the same as an
*idempotent*$A\to A$ of $\mathcal{C}$, where $\mathbb{B}=(\{0,1\},\text{OR},1)$.

**The $\infty$-categorical setting.**

**Preliminary Question.** Given an $\infty$-category $\mathcal{C}$, is there a natural monoidal $\infty$-groupoid structure on $\mathrm{Hom}_{\mathcal{C}}(X,X)$?

**Question.** Is there an analogue of deloopings for $(\infty,1)$-categories, where we start with a monoidal $\infty$-groupoid $\mathcal{C}$ and construct an $(\infty,1)$-category $\mathbf{B}\mathcal{C}$ such that

- A functor $\mathbf{B}\mathcal{C}\to\mathcal{D}$ from $\mathbf{B}\mathcal{C}$ to another $(\infty,1)$-category $\mathcal{D}$;

is the same thing as

- An object of $\mathcal{D}$ together with a functor of monoidal (?) $\infty$-groupoids $\mathcal{C}\to\mathrm{Hom}_{\mathcal{D}}(X,X)$,

with $\mathrm{Hom}_{\mathcal{D}}(X,X)$ the morphism space of $\mathcal{D}$ from $X$ to itself, and where this bijection can be made into a full-fledged isomorphism of (appropriate) $\infty$-groupoids?

Delooping monoidal $(\infty,1)$-categories into $(\infty,2)$-categories. $\endgroup$