A simplicial group is, naturally enough, a simplicial object in the category of groups. It is equivalently a simplicial set {G_\bullet} such that each {G_n} has a group structure and all the face and degeneracy maps are group homomorphisms. By a well-known lemma, a simplicial group is automatically a Kan complex. We can thus think of a simplicial group in two ways. On the one hand, it is an {\infty}-groupoid, as a Kan complex. On the other hand, we can think of a simplicial group as a model for a “higher group,” or an {\infty}-group. This is the intuition that the nLab suggests.

Now it is well-known that in ordinary category theory, a group is the same as a groupoid with one object. So an {\infty}-groupoid with one object should be the same as an {\infty}-group. This is in fact true with the above notation. In other words, if we say that an {\infty}-groupoid is a Kan complex (as usual), and decide that an {\infty}-group is a simplicial group, then the {\infty}-groups are the “same” as the {\infty}-groupoids with one object.

Here the “same” means that the associated {\infty}-categories are equivalent. One way of expressing this is to say that there are natural model structures on {\infty}-groupoids with one object and simplicial groups, and that these are Quillen equivalent. This is a frequent way of expressing the idea that two {\infty}-categories (or at least, {(\infty, 1)}-categories) are equivalent. For instance, this is the way the {\infty}-categorical Grothendieck construction is stated in HTT, as a Quillen equivalence between model categories of “right fibrations” and simplicial presheaves. Let us express this formally.

Theorem 1 (Kan) There are natural model structures on the category {\mathbf{SGrp}} of simplicial groups and on the category {\mathbf{SSet}_0} of reduced simplicial sets (i.e. those with one vertex), and the two are Quillen equivalent.

 

To give this construction, we will first describe the classifying space of a simplicial group. One incarnation of this is going to give the functor from simplicial groups to reduced simplicial sets. This is a simplicial version of the usual topological classifying space. Recall that if {G} is a topological group, then there is a classifying space {BG} and a principal {G}-bundle {EG \rightarrow BG} such that {EG} is contractible. It follows from this that for any CW-complex {X}, the homotopy classes of maps {X \rightarrow BG} are in bijection with the principal {G}-bundles on {X} (in fact, {EG \rightarrow BG} is a universal bundle).

We will need the appropriate notion of a principal bundle in the simplicial setting. Let {G_\bullet} be a simplicial group.

Definition 2 A {G_\bullet}–simplicial set is a simplicial set {X_\bullet} together with an action {G_\bullet \times X_\bullet \rightarrow X_\bullet} satisfying the usual axioms. Thus, each {X_n} is a {G_n}-set. There is a category {\mathbf{SSet}_G} of {G_\bullet}-simplicial sets and {G_\bullet}-equivariant maps.

We next give the definition of a principal bundle. Notice that because of the combinatorial nature of simplicial sets, we don’t make any local trivialty condition; that will fall out.

 

Definition 3 Let {E_\bullet \rightarrow B_\bullet} be a map of {G_\bullet}-simplicial sets. We say that {E_\bullet \rightarrow B_\bullet} is a principal {G_\bullet}-bundle if {B_\bullet} has trivial action, each {E_n} is a free {G_n}-set, and if {E_\bullet/G_\bullet \simeq B_\bullet} under the natural map. (more…)