How equivalent are the theories of reduced and groupal $\infty$-groupoids?

Question

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?

Answer 1

To answer your first set of questions in order (I'm going to use the word "space" for "$\infty$-groupoid"):

Yes, this is all correct.

You seem to be familiar with how to construct $BG$ as a simplicial space via a bar construction. To turn this into an actual space, you just have to form the geometric realization of this simplicial space. This is just like the geometric realization of a simplicial set: if $X_*$ is a simplicial space, its realization is the quotient of $\bigsqcup X_n \times \Delta^n$ given by gluing together simplices according to the face (and degeneracy) maps. More succinctly, it is the coend of the functor $X:\Delta^{op}\to Spaces$ along the functor $\Delta \to Spaces$ sending $[n]$ to the standard $n$-simplex. More abstractly, it is the (homotopy) colimit of the diagram $X:\Delta^{op}\to Spaces$ in the $(\infty,1)$-category of spaces. If you model spaces as simplicial sets, it can also be described as the pullback of your bisimplicial set along the diagonal functor $\Delta\to\Delta^2$ (to show this, you need to only verify it for single "bisimplices" (ie, representable functors on $\Delta^2$), which amounts to the fact that the product of two representable functors on a category $C$ is the pullback of the associated representable functor on $C^2$ along the diagonal $C\to C^2$).

I don't know the best way of precisely formulating and proving this, but one way is to construct an explicit Quillen equivalence between the categories of simplicial groups and the category of reduced simplicial sets (simplicial groups can be used instead of $A_\infty$-groups because $A_\infty$-spaces can be rigidified). For this, the delooping functor is exactly just taking the geometric realization of the bar construction. The looping functor is subtler--one has to be careful to get a functor which actually lands in simplicial groups. Even if you were happy to land in $A_\infty$-groups, just taking the simplicial mapping space $Maps(S^1,X)$ would not work because for non-fibrant objects $X$ this does not have a multiplication. The classical solution to this is called the "Kan loop group", the exact details of which I don't remember but are described on nlab in the generalized setting of a not-necessarily-reduced simplicial set (in which case you get a simplicial groupoid, rather than a simplicial group). In any case, this is just a specific simplicial model for the loopspace of a reduced simplicial set which happens to actually be a group.

As for a reference, I learned the Kan loop group and the Quillen equivalence it gives from Goerss-Jardine's book Simplicial Homotopy Theory, but they say essentially nothing about thinking about this from an $\infty$-categorical perspective.