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

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?

• Commented Sep 18, 2021 at 2:26
• If you think of Infinity categories as simplicial categories and Infinity groupoids as simplicial sets you'll realize that the answer is clearly 'yes' Commented Sep 18, 2021 at 9:30
• @FernandoMuro To be fair this argument passes through a non-trivial strictification theorem (that every $E_1$-space can be represented by a simplicial monoid) Commented Sep 18, 2021 at 9:31

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.
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.