Delooping monoidal $\infty$-groupoids into $\infty$-categories

Question

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?

Answer 1

I assume that with ``monoidal ∞-groupoid'' you mean an $E_1$-space. In this case the answer is yes. It is well known that $E_1$-spaces can be modeled by functors

$$X:\Delta^{\mathrm{op}}\to \operatorname{Space}$$ satisfying the Segal conditions. Now if you are given an $\infty$-category $\mathcal{C}$ you can define a simplicial space $$s(\mathcal{C}): [n]\mapsto\operatorname{Map}_{\operatorname{Cat}_∞}(\Delta^n,\mathcal{C})\,.$$ In fact this functor is fully faithful and identifies $\operatorname{Cat}_∞$ with the category of complete Segal spaces. For the following we won't need all this though - we will use only that it takes values in Segal spaces (which follows immediately from $\Delta^n\amalg_{\Delta^0} \Delta^m\simeq\Delta^{n+m-1}$ in $\operatorname{Cat}_∞$).

Now let $x\in\mathcal{C}$ be an object of $\mathcal{C}$. Then we can define the simplicial space $$ \operatorname{End}_{\mathcal{C}}(x):\Delta^{\mathrm{op}}\to \operatorname{Space}\qquad [n]\mapsto \{x\}\times_{\operatorname{Map}(\{0,\dots,n\},\mathcal{C})} \operatorname{Map}_{\operatorname{Cat}_∞}(\Delta^n,\mathcal{C})\,.$$ That is it sends $[n]$ to the (∞-)groupoid of functors $F:\Delta^n\to \mathcal{C}$ that sends all objects to $x$. It is easy now to see that $\operatorname{End}_{\mathcal{C}}(x)$ satisfies the Segal conditions and so it is an $E_1$-space.

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This takes care of your preliminary question. To go back to your main question, the functor $(\mathcal{C},x)\mapsto \operatorname{End}_{\mathcal{C}}(x)$ obviously preserves all limits and filtered colimits, and so it has a left adjoint $B$ exactly as you wanted. To get a more ``concrete'' description $B$ sends an $E_1$-space $X$ to the ∞-category corresponding to the completion of $X$ seen as a Segal space. That is $$BX:=\int^{[n]\in\Delta^{\mathrm{op}}} X([n])\times \Delta^n$$ where the coend is computed in $\operatorname{Cat}_∞$.

With more care one can show that $B:E_1-\operatorname{Space}\to(\operatorname{Cat}_∞)_{\Delta^0/}$ is fully faithful with essential images those arrows $\Delta^0\to\mathcal{C}$ that are essentially surjective (that is such that $\mathcal{C}$ has only one equivalence class of objects). Indeed this is a special case of the equivalence between the ∞-category of Segal spaces and the ∞-category of flagged ∞-categories.