I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this question is that my category number has been hovering somewhere around $2$, and I'd like to increase it, but $\infty$ is often easier than $3$.

Suppose that I have some familiarity with the following words (meaning, feel free to "remind" me what the correct definitions are):

- Some version (Stasheff associahedra?) of
*$A_\infty$ monoids*. - Kan simplicial sets as
*$\infty$-groupoids*.

One can then define the following: A *groupal $\infty$-groupoid* is an $A_\infty$ monoid $G$ in $\infty$-groupoids, such that the map $(g,h) \mapsto (g,gh)$, $G \times G \to G\times G$ is an equivalence of $\infty$-groupoids. (If this isn't quite the right definition, please let me know.)

One could instead talk about $\infty$-groupoids for which the set of $0$-morphisms is a point. I think this is what are called *reduced*.

I'm under the impression that these should be "the same". If I were working not with $\infty$-groupoids but rather at a low categorical level, I would understand how they are the same: a *groupoid with one object* is "the same" as a *group* or *groupal set* (an associative monoid such that the map $(g,h) \mapsto (g,gh)$ is an isomorphism).

More precisely, there should be functors $\Omega$ and $\rm B$ between the $(\infty,1)$-categories of reduced $\infty$-groupoids and groupal $\infty$-groupoids, and I would assume that these are an equivalence, in the appropriate sense.

I almost understand these functors:

- Given a reduced $\infty$-groupoid $G$, I would try to define $\Omega G = \hom(S,G)$, where $S$ is some $\infty$-groupoid version of the circle, say a particular $S = \mathrm B\mathbb Z$ that I might construct by hand. The "$\hom$" is just the hom of $\infty$-groupoids (reduced $\infty$-groupoids are full in $\infty$-groupoids), and in particular it takes values in $\infty$-groupoids; on the other hand, letting $\vee$ denote the coproduct in reduced groupoids, there is a distinguished map $S \to S\vee S$ which winds around the outside of the figure-eight, and pulling back along this map gives the groupal structure on $\Omega G$. Left to check is that this really is a groupal structure, but that should be easy.
- Given a groupal $\infty$-groupoid $G$, I should try to define $\mathrm B G$ in the same way that I would if I were just starting with a group. But a priori I only see how to define $\mathrm B G$ as a simplicial object in $\infty$-groupoids. So my biggest difficulty here is that don't know how to collapse what I'm modeling as a "double simplicial set" into a "single simplicial set". Writing $\Delta$ for the category whose objects are finite totally-ordered sets and whose morphisms are non-decreasing maps (so that a simplicial set is a functor $\Delta^{\mathrm{op}} \to \mathrm{Set}$), maybe there is a nice map $\Delta \to \Delta^{\times 2}$ along which I can pull back? If so, then there only remains to check the Kan condition.
- Oh, and I'd need to check that $\Omega,\mathrm B$ are inverse (up to ...) to each other.

After a rambly introduction, my questions are:

Is this all correct? What is the $\mathrm B$ construction? What's the precise statement of the equivalence between groupal and reduced $\infty$-groupoids?

I assume that this type of thing is carefully spelled out somewhere in the literature. So maybe my real question is:

What is a good reference that will take my hand and walk me through this part of category theory?